Phase transition between the quantum spin Hall and insulator phases in 3Demergence of a topological gapless phase

For most of physics history, the assumption ran like this: a system is one phase or it is another. You tune a parameter, the gap closes at a single knife-edge point, and the material flips from one topological class to its opposite. That assumption held cleanly in two dimensions. Then Shuichi Murakami showed it breaks down in three dimensions — and in the break, something genuinely new appears. Between the quantum spin Hall phase and the ordinary insulator, in three-dimensional crystals that lack inversion symmetry, there is a third thing: an extended, stable, topologically protected gapless phase that lives not at a point but across a whole corridor of parameter space. To understand why that's extraordinary, you have to start with the two phases it sits between. Both an ordinary band insulator and a quantum spin Hall insulator, or QSH for short, have a gap in their bulk energy spectrum. Neither conducts electricity through the interior. By that measure, they look identical. But they are distinguished by a topological label called the Z2 invariant, written as nu, which can take only two values: even or odd. If nu is odd, the system is in the QSH phase. If even, it's ordinary. The Z2 number answers a qualitative question about the electron wavefunctions: do they carry a topological twist that forces conducting states to exist at the surface, no matter what? The physical mechanism behind the QSH phase is spin-orbit coupling, or the interaction between an electron's spin and its orbital motion through a crystal. A useful picture is two quantum Hall systems running simultaneously, one for spin-up electrons and one for spin-down, with opposite effective magnetic fields. The up-spin Hall conductance equals e squared over h; the down-spin Hall conductance equals minus e squared over h. At the boundary, you get counterpropagating edge states: spin-up moves one way, spin-down moves the other. These form what are called Kramers pairs, which are protected by time-reversal symmetry. Nonmagnetic disorder can't gap them out because time-reversal symmetry won't allow it. In three dimensions, the topology is richer. It's described by four Z2 numbers written as nu zero followed by nu one, nu two, and nu three. Only nu zero is fully robust against weak disorder; the others require relatively clean samples. For crystals with inversion symmetry, computing these invariants simplifies dramatically: you take the parity eigenvalues of occupied Kramers pairs at the special high-symmetry points in momentum space and multiply them together. That shortcut matters enormously for materials searches, but we'll come back to it. First, let's discuss the baseline picture in two dimensions, because Murakami uses it as the contrast. In two dimensions, when you tune a control parameter to drive a transition between QSH and ordinary insulating phases, the gap must close. It closes at a single, discrete parameter value — a knife-edge — and reopens on the other side with the Z2 invariant flipped. Think of untying a knot: there is exactly one moment of slack you must pass through. The transition is sharp. There is no room to be neither one phase nor the other. This is the two-dimensional picture Murakami is working against. In three dimensions, for inversion-asymmetric systems — crystals without a center of symmetry — something different happens. As you increase a control parameter m, the bulk gap closes at m equals m one, the system becomes gapless, remains gapless as m increases, and only reopens the gap at m equals m two. The gapless region between m one and m two is not a blurry transition. It is a distinct thermodynamic phase. Why does this happen? The answer lies in how many conditions must be simultaneously satisfied to make two bands touch. For two non-degenerate bands in an inversion-asymmetric crystal, a crossing requires satisfying three independent equations: the two diagonal elements of a two-by-two Hamiltonian must be equal, and both the real and imaginary parts of the off-diagonal element must vanish. Three equations in a four-dimensional space — three momenta plus the control parameter m — generically produce solutions along a one-dimensional curve. A line of gapless points, not a single isolated crossing. That curve is what sweeps through parameter space as m is tuned from m one to m two. Now for the mechanism that keeps this phase stable, and this is where the topology becomes vivid. Murakami identifies each band-touching point — each diabolical point — as a monopole in momentum space, carrying an integer topological charge. For a linear, Weyl-like band crossing, that charge is plus or minus one. And just like magnetic monopoles, they cannot vanish alone. A monopole can only be annihilated by meeting an antimonopole of opposite charge. The corrected picture Murakami gives in the corrigendum makes the timeline explicit. At m equals m one, monopole-antimonopole pairs — charge plus one and charge minus one — are created in pairs at specific high-symmetry points in momentum space. Time-reversal symmetry forces the configuration to remain symmetric about the point k equals G over two throughout their migration. The monopoles and antimonopoles move through momentum space as m increases, staying symmetric, unable to disappear. Only at m equals m two, when they meet again at the high-symmetry points k equals G over two, do they annihilate pairwise and allow the bulk gap to reopen. The gapless phase lives entirely inside that corridor between creation and annihilation. In two dimensions, the analogous diabolical points carry no such conserved charge. A band touching in two dimensions can be removed by a generic small perturbation — there is no topological protection, no monopole to conserve. That is precisely why two dimensions has no intermediate gapless phase. The distinction is purely geometric: in three dimensions, a gapless point is a topological object; in two dimensions, it is not. Murakami also addresses stability against disorder and electron-electron interactions directly. He invokes a Laughlin-type flux argument: even in a disordered system, you can thread magnetic fluxes through a ring geometry and define the Z2 invariant in terms of those fluxes rather than Bloch momenta. As long as interactions and disorder are weak enough not to close the bulk gap of the surrounding gapped phases, the Z2 number remains well-defined and identical to the clean, noninteracting value. The topological gapless phase — protected by the need for the Z2 invariant to jump between two distinct values on either side — survives weak perturbations because its stability is geometric, not energetic. Murakami then turns to materials. He outlines two practical routes to computing the Z2 invariant for real crystals. The Pfaffian method works in general but is numerically expensive; it requires tracking the Pfaffian of a matrix of Bloch wavefunctions across a discretized momentum space and counting vortices. The parity method, available when the crystal has inversion symmetry, is far simpler: at each of the time-reversal-invariant momenta, take the product of parity eigenvalues of all occupied Kramers pairs to get a quantity delta. Then the Z2 parity is the product of all those delta values at the relevant special k points. For physical criteria, Murakami points to strong spin-orbit coupling combined with a true bulk gap. Bismuth is the paradigm case. Films thinner than roughly 30 nanometres were expected theoretically to become insulating, but angle-resolved photoemission spectroscopy measurements on ten-bilayer films remained semimetallic — evidence that the problem is subtle in practice. The alloy bismuth-antimony, Bi one minus x Sb x, with x between 0.07 and 0.22, is proposed as a window where the right gap opens. Numerical studies modulating interlayer hopping in bismuth and antimony reveal explicit phase boundaries: for bismuth, a hopping parameter f spanning zero to zero point two two three, zero point two two three to zero point nine nine three, and zero point nine nine three to one corresponds to three distinct topological classes; for antimony, two boundaries split the range at f equals zero point five four. These numbers matter because they show the phase boundaries are not theoretical abstractions — they are tunable, calculable, and accessible in principle to experiment. The deeper implication of Murakami's work is that topological protection need not be confined to gapped phases. The monopole structure of momentum space can stabilize a gapless phase just as firmly as it stabilizes a gapped one — and when it does, that gapless phase is not a transition but a destination. Between two insulators that differ in their topological label, there can be a corridor that belongs to neither, protected by geometry, robust against disorder, and visible only if you know to look in three dimensions. The universe of possible phases is wider than a binary, and sometimes the most interesting physics lives exactly in between. This lecture was created by ennepō. Go to https://ennepo.ai to Discover, Create and Follow the latest research in your field. Read when you can. Listen when you want to.