A topologically ordered material is characterized by a rare quantum organization of electrons that evades the conventional spontaneously broken symmetry–based classification of condensed matter. Exotic spin-transport phenomena, such as the dissipationless quantum spin Hall effect, have been speculated to originate from a topological order whose identification requires a spin-sensitive measurement, which does not exist to this date in any system. Using Mott polarimetry, we probed the spin degrees of freedom and demonstrated that topological quantum numbers are completely determined from spin texture–imaging measurements. Applying this method to Sb and Bi _1–x Sb _x , we identified the origin of its topological order and unusual chiral properties. These results taken together constitute the first observation of surface electrons collectively carrying a topological quantum Berry's phase and definite spin chirality, which are the key electronic properties component for realizing topological quantum computing bits with intrinsic spin Hall–like topological phenomena.

Lifted Kramers spin degeneracy (LKSD) has been among the central topics of condensed-matter physics since the dawn of the band theory of solids^1,2. It underpins established practical applications as well as current frontier research, ranging from magnetic-memory technology^3–7 to topological quantum matter^8–14. Traditionally, LKSD has been considered to originate from two possible internal symmetry-breaking mechanisms. The first refers to time-reversal symmetry breaking by magnetization of ferromagnets and tends to be strong because of the non-relativistic exchange origin¹⁵. The second applies to crystals with broken inversion symmetry and tends to be comparatively weaker, as it originates from the relativistic spin–orbit coupling (SOC)^16–19. A recent theory work based on spin-symmetry classification has identified an unconventional magnetic phase, dubbed altermagnetic^20,21, that allows for LKSD without net magnetization and inversion-symmetry breaking. Here we provide the confirmation using photoemission spectroscopy and ab initio calculations. We identify two distinct unconventional mechanisms of LKSD generated by the altermagnetic phase of centrosymmetric MnTe with vanishing net magnetization^20–23. Our observation of the altermagnetic LKSD can have broad consequences in magnetism. It motivates exploration and exploitation of the unconventional nature of this magnetic phase in an extended family of materials, ranging from insulators and semiconductors to metals and superconductors^20,21, that have been either identified recently or perceived for many decades as conventional antiferromagnets^21,24,25.

We show that the strongly spin-orbit coupled materials ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ and ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ and their derivatives belong to the $Z_2$ topological-insulator class. Using a combination of first-principles theoretical calculations and photoemission spectroscopy, we directly show that ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ is a large spin-orbit-induced indirect bulk band gap ($\delta \sim 150\mathrm{meV}$) semiconductor whose surface is characterized by a single topological spin-Dirac cone. The electronic structure of self-doped ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ exhibits similar $Z_2$ topological properties. We demonstrate that the dynamics of spin-Dirac fermions can be controlled through systematic Mn doping, making these materials classes potentially suitable for topological device applications.

A topological insulator protected by time-reversal symmetry is realized via spin–orbit interaction-driven band inversion. The topological phase in the Bi_1−xSb_x system is due to an odd number of band inversions. A related spin–orbit system, the Pb_1−xSn_xTe, has long been known to contain an even number of inversions based on band theory. Here we experimentally investigate the possibility of a mirror symmetry-protected topological crystalline insulator phase in the Pb_1−xSn_xTe class of materials that has been theoretically predicted to exist in its end compound SnTe. Our experimental results show that at a finite Pb composition above the topological inversion phase transition, the surface exhibits even number of spin-polarized Dirac cone states revealing mirror-protected topological order distinct from that observed in Bi_1−xSb_x. Our observation of the spin-polarized Dirac surface states in the inverted Pb_1−xSn_xTe and their absence in the non-inverted compounds related via a topological phase transition provide the experimental groundwork for opening the research on novel topological order in quantum devices.

We report a Rashba spin splitting of a two-dimensional electron gas in the topological insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ from angle-resolved photoemission spectroscopy. We further demonstrate its electrostatic control, and show that spin splittings can be achieved which are at least an order-of-magnitude larger than in other semiconductors. Together these results show promise for the miniaturization of spintronic devices to the nanoscale and their operation at room temperature.