Phillips reveals that ‘1/4 is the new 1/2’ when electron interactions in topological insulators are considered

New work has overturned long-held beliefs that were based on overly simplistic assumptions.

For decades, condensed matter physicists have studied certain topological problems under the assumption that electrons in a system being studied don’t interact with each other—an unrealistic assumption that was considered necessary so that calculations would be simple enough to solve exactly.

Now, physics professor Philip W. Phillips and his colleagues have revealed not only that it is possible to take electrons’ interactions into consideration, but that unexpected phenomena are found when they are. Specifically, the electrons acquire a one-way current at the edge of a sample material when the total electron filling in the system is 1/4, rather than the 1/2 filling that was expected when it was assumed that electrons did not interact.

What does that mean? Phillips said that until now, “topology was viewed as possible only in a half-filled system: you have a lower band, you have an upper band; the electrons have an energy that’s in between them; the lower band is all full; the top one is empty. That’s the setup for a topological insulator. Now we’re saying the picture is much more complicated in the presence of strong electron-electron interactions, and it enhances the number of ways now that you can make a topological insulator.”

What’s that more complicated picture? Phillips explained that when his team’s calculations accounted for the way electrons naturally repel each other, they discovered that those forces cause the formation of a second pair of conduction and valance bands, for a total of four bands.

“All of the topology which people associated with [the original two-band understanding] is still present, but now it’s at a completely different filling,” said Phillips. The topology is now seen to be present first at 1/4 filling and then at 3/4 filling, for example.

Phillips says that researchers have long sought to expand the range of topological states of matter, and that the new findings reveal that consideration of electrons’ actual behavior will be key to doing so.

What’s the significance of that one-way current that his team discovered? The paradigmatic example of such one-way, or “chiral,” currents (see the figure, part a) is the quantum Hall effect (QHE), which requires the application of a perpendicular magnetic field. Physicists once pondered whether such a current could occur without a perpendicular magnetic field, and the answer turned out to be a resounding yes—a realization that ushered in the field of topological insulators.

Two phenomena emerged in that context: the quantum anomalous Hall (QAH) and quantum spin Hall (QSH) effects (see figure, parts b and c), neither of which involves an external magnetic field. The models that originally yielded such physics all assumed non-interacting electrons and 1/2 filling; further, according to the old understanding, it would be impossible ever to observe both effects in the same sample.

The new findings emerged from research that addressed the question: Can those same physics transpire at any other band filling?

As Phillips put it, “The only other knob at play is the interactions between the electrons.”

By pursuing that “knob,” he and his colleagues found that although the QAH and QSH effects are quite different (see figure), strong interactions allow them to exist in the same sample—something never before recognized—and that transitions between the two effects can be triggered simply by raising or lowering the temperature.

Specifically, they found that at a low temperature, only one of the two currents found in QSH survives, giving rise to the one-way current found in QAH. The twist is that the system filling must now be 1/4, not the 1/2 expected in the non-interacting system.

“This is quite a surprise,” Phillips said, “as in the non-interacting system, no sign of topology is present at 1/4 filling or, in fact, [at] any odd multiple of 1/4, for example 3/4.”

Phillips’s co-author Benjamin E. Feldman, of Stanford University, has now made some unprecedented laboratory observations of the 1/4 state, confirming the phenomena predicted both by the team’s initial pen-and-paper proof and by their subsequent computer simulation.

“It was gratifying to see our prediction observed experimentally,” said Phillips.

The work shows that interactions offer a new window into topological states of matter that non-interacting electron physics could not have foreseen. Now, the discovery of other fractional filling states, free of the complication of a magnetic field, is a real possibility.

As Phillips put it, “Move over free-electron physics; a new era of interactions has sprung!”

Details of the work can be found in a new Nature Communications paper by Phillips, postdoc Peizhi Mai, graduate student Jinchao Zhao, and Feldman, entitled “1/4 is the new 1/2 when topology is intertwined with Mottness.” A description of the experimental results is available in an arXiv preprint prepared by Feldman and his colleagues; Fig. 1(d–g) is of particular interest.

This research was supported by the Center for Quantum Sensing and Quantum Materials, a DOE Energy Frontier Research Center, under Grant No. DE-SC0021238; and by the National Science Foundation under Grant No. DMR-2111379. The findings presented are those of the researchers and not necessarily those of the funding agencies.