Unconventional transformation of spin Dirac phase across a topological quantum phase transition

The topology of a topological material can be encoded in its surface states. These surface states can only be removed by a bulk topological quantum phase transition into a trivial phase. Here we use photoemission spectroscopy to image the formation of protected surface states in a topological insulator as we chemically tune the system through a topological transition. Surprisingly, we discover an exotic spin-momentum locked, gapped surface state in the trivial phase that shares many important properties with the actual topological surface state in anticipation of the change of topology. Using a spin-resolved measurement, we show that apart from a surface bandgap these states develop spin textures similar to the topological surface states well before the transition. Our results offer a general paradigm for understanding how surface states in topological phases arise from a quantum phase transition and are suggestive for the future realization of Weyl arcs, condensed matter supersymmetry and other fascinating phenomena in the vicinity of a quantum criticality.

intensity fluctuation, which is proportional to the variation of the elemental composition as a function of real-space location, is found to be within ±2%. We note that the typical size of the ARPES beamspot ranges from 200 to 500 µm 2 , whereas the EDS measurements have a spatial resolution of about 100 nm 2 . Thus these measurements rule out the possibility that our results are artifacts due to spatial imhomogeneity of the samples. We note that the n-doping in the samples are caused by Se vacancies.
And depending on the number of vacancies in a crystal, its chemical potential will slightly vary from batch to batch. We do notice a small variation of the chemical potential ( Fig. 1b of the main text).
In addition to the EDS measurements, we have also performed ARPES measurements on different locations of the same cleaved surface of each sample studied, in order to check the in-situ spatial homogeneity in our ARPES experiments. Supplementary are obtained, which supports that the ARPES results are not a spatial superposition of different compositions. Such in-situ spatial homogeneity of the ARPES spectra is a good indicator of a homogeneous sample. Otherwise one would have to have a "uniform" spatial inhomogeneity, i.e. a phase/stoichiometric inhomogeneity within the spot size that is uniform across the whole sample.
• Incident photon energy dependence studies and k z dispersion measurements • A phenomenological picture for the preformed surface states In order to better understand the spin texture of the quasi-2D states, we propose a phenomenological picture consistent with the basic topological physics for our observation: As shown in Supplementary Figures 7,8, the quasi-2D states can be viewed as a Rashba-like state, whose inner band is not observable because it is severely damped due to its strong overlap with the bulk bands in E − k space (see Supplementary Figures 6,7 for a schematic). As the system is tuned approaching the TCP from the trivial side, the inner band completely loses its surface character, whereas the outer band is systematically enhanced in terms of its surface spectral weight and spin polarization, and evolve into the topological surface states (as clearly observed in our data). We emphasize that we use the term "Rashba-like" for the observed perform surface states because there are two singly degenerate bands as in a real Rashba 2DEG. However, the Rashba surface states are due to a combined effect of atomic spin-orbit coupling and the electrical field perpendicular to the surface, and follows the Rashba Hamiltonian, whereas it is not fully applicable for the observed preformed surface states. This issue needs further theoretical studies to illuminate the microscopic origin of the preformed surface states in theory.
• Observation of preformed surface states in another topological phase transition system (Bi 1−δ In δ ) 2 Se 3 .
Here we present ARPES and spin-resolved ARPES data, which show our obser-

• EDS measurements
In an EDS experiment, a high-energy electron beam is focused onto the surface of the sample being studied. At rest, an atom within the sample contains ground state (or unexcited) electrons in discrete energy levels or electron shells bound to the nucleus.
The incident electron beam excites an electron in an inner shell (K, L, or M), ejecting it from the shell while creating a core-hole where the electron was. An electron from an outer, higher-energy shell then fills the hole, and the difference in energy between the higher-energy shell and the lower energy shell may be released in the form of an x-ray with a unique energy level characterizing a certain element. The intensity and energy of the x-rays emitted from a sample can be measured by an energy-dispersive spectrometer.
For our EDS instrument, the incident electron beam is at energy of 15 keV. The emitted x-ray photons are detected by an EDS crystal detector. Our EDS detector contains a silicon crystal that absorbs the energy of incoming x-rays by ionization, yielding free electrons in the crystal that become conductive and produce an electrical charge bias. The x-ray absorption thus converts the energy of individual x-rays into electrical voltages of proportional size; the electrical pulses correspond to the characteristic xrays of the element. The typical energy resolution of our EDS crystal detector is about 100 eV (note that the typical core-level emitted x-ray energy is much higher, on the order of 1000eV to 10000 eV). The spatial resolution is defined by the mean free path of the incident electron beam, which is typically around 100 nm 2 . This enables us to resolve the relative chemical composition in real space with a spatial resolution of 100 nm 2 .

• Model theoretical calculations
We have created a Green's function implementation of the experimentally-based k · p model in Refs. [1,2] to simulate the bulk and surface dispersions of BiTl(S 1−δ Se δ ) 2 .
Any solid with a surface can be described as a semi-infinite chain of principal layers with nearest-neighbor interactions [3]. In case of BiTl(S 1−δ Se δ ) 2 each unit cell represents one principal layer and the effective Hamiltonian for the unit cell is given by where v is the Fermi velocity, m 1 and m 2 are the orbital masses and the parameter d is introduced to generate a gap [1]. The above Hamiltonian is constructed considering two equivalent Se/S atoms (in one rhombohedral unit cell) where each Se/S atom has two p z orbitals, one with up spin and the other with down spin. The hopping between two adjacent unit cells is realized by where t z is the nearest neighbor hopping parameter. To illustrate the single-Dirac-cone topological surface states, the chosen parameters are : with the surface site at 1. For the system under study, H 1 = H 2 = H 3 = H p and T 1 = T 2 = T . H p and T are given in Eq. S1 and S2, respectively. A renormalization approach is used to evaluate the Green's function (GF) of this chain [4,5]. In this approach alternate sites on the chain are eliminated and the equations for the GF are used to define a new effective Hamiltonian with renormalized diagonal elements for the remaining sites and renormalized interactions between the remaining adjacent sites. This decimation process is repeated iteratively until the effective interaction between remaining adjacent sites is as small as one wishes. Thus, the renormalized Hamiltonian for the surface site is and for the bulk site is The renormalized coupling between the layers is The Green's functions for the surface and bulk are obtained by inverting ω −H s and ω −H b , respectively. Now we also want to study the effect of external potential on the surface electron kinetics. When an extra potential is added at the sample surface, the surface layer and a few layers near the surface are effected, since the potential decreases rapidly as we move away from the surface towards the bulk. This effect is simulated by adding more layers on the original surface, where each layer has an appropriate potential.
After adding one layer the new Hamiltonian is given bȳ where V 0 is the potential on the additional layer and H 0 = H p .
The Green's function G ′ for the new system is calculated by using the GF for the old system. The site-diagonal GF for the new surface layer at site 0 is given by 17 and the site-diagonal GF at site 1 becomes Using this method, extra layers can be added iteratively to the chain.