## Abstract

The major breakthroughs in understanding of topological materials over the past decade were all triggered by the discovery of the Z_{2}-type topological insulator—a type of material that is insulating in its interior but allows electron flow on its surface. In three dimensions, a topological insulator is classified as either ‘strong’ or ‘weak’^{1,2}, and experimental confirmations of the strong topological insulator rapidly followed theoretical predictions^{3,4,5}. By contrast, the weak topological insulator (WTI) has so far eluded experimental verification, because the topological surface states emerge only on particular side surfaces, which are typically undetectable in real three-dimensional crystals^{6,7,8,9,10}. Here we provide experimental evidence for the WTI state in a bismuth iodide, β-Bi_{4}I_{4}. Notably, the crystal has naturally cleavable top and side planes—stacked via van der Waals forces—which have long been desirable for the experimental realization of the WTI state^{11,12}. As a definitive signature of this state, we find a quasi-one-dimensional Dirac topological surface state at the side surface (the (100) plane), while the top surface (the (001) plane) is topologically dark with an absence of topological surface states. We also find that a crystal transition from the β-phase to the α-phase drives a topological phase transition from a nontrivial WTI to a normal insulator at roughly room temperature. The weak topological phase—viewed as quantum spin Hall insulators stacked three-dimensionally^{13,14}—will lay a foundation for technology that benefits from highly directional, dense spin currents that are protected against backscattering.

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## Data availability

The data that support the findings of this study are available from the corresponding authors on reasonable request.

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## Acknowledgements

We thank Y. Okada, D. Hamane, M. Lippmaa, Y. Yoshida, T. Miyamachi, T. Hattori, Y. Hasegawa and F. Komori for scanning electron microscope (SEM)/atomic force microscope (AFM)/scanning tunnelling microscope (STM) characterization of the Bi_{4}I_{4} surface and for fruitful discussions; and Y. Ishida for supporting analysis of data. We thank the Diamond Light Source for access to beamline I05 under proposals SI15095, SI16161 and SI17816, contributing to the results presented here. We also thank the Elettra Light Source for access to the spectromicroscopy beamline. The GISAXS experiments were performed under the approval of Public Finance–Public Accountability Collective (PF–PAC) no. 2016G548. The work done at the Tokyo Institute of Technology was supported by a JST Core Research for Evolutional Science and Technology (CREST) project (JPMJCR16F2) and a Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for Scientific Research (B) (JP16H03847). R.N. acknowledges support from JSPS under KAKENHI grant JP18J21892, and by JSPS through the Program for Leading Graduate Schools (Advanced Leading Course for Photon Science). This work was also supported by the ‘Topological Materials Science’ (JP16H00979) KAKENHI on Innovative Areas from JSPS, and by JSPS KAKENHI grants JP16H06013, JP17K14319, JP16H02209, JP16K13829, and JP18H01165. R.A. acknowledges support from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, under KAKENHI grant JP16H06345.

### Reviewer information

*Nature* thanks J. Sanchez-Bárriga and the other anonymous reviewer(s) for their contribution to the peer review of this work.

## Author information

### Affiliations

#### Institute for Solid State Physics, University of Tokyo, Kashiwa, Japan

- Ryo Noguchi
- , K. Kuroda
- , M. Sakano
- , C. Bareille
- , M. Nakayama
- , K. Yaji
- , A. Harasawa
- , S. Shin
- & Takeshi Kondo

#### Materials and Structures Laboratory, Tokyo Institute of Technology, Yokohama, Japan

- T. Takahashi
- & T. Sasagawa

#### Department of Physics, Osaka University, Toyonaka, Japan

- M. Ochi

#### National Metrology Institute of Japan, National Institute of Advanced Industrial Science and Technology, Tsukuba, Japan

- T. Shirasawa

#### Department of Applied Physics and Quantum-Phase Electronics Center (QPEC), The University of Tokyo, Tokyo, Japan

- M. Sakano

#### Diamond Light Source, Harwell Campus, Didcot, UK

- M. D. Watson
- , H. Iwasawa
- , P. Dudin
- , T. K. Kim
- & M. Hoesch

#### Graduate School of Science, Hiroshima University, Higashi-Hiroshima, Japan

- H. Iwasawa

#### DESY Photon Science, Deutsches Elektronen-Synchrotron, Hamburg, Germany

- M. Hoesch

#### Elettra - Sincrotrone Trieste, Basovizza, Italy

- V. Kandyba
- , A. Giampietri
- & A. Barinov

#### RIKEN Center for Emergent Matter Science (CEMS), Wako, Japan

- R. Arita

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### Contributions

T.K. and T.Sa. planned the experimental project. R.N. and K.K. conducted the ARPES experiment and analysed the data. M.S., C.B., M.N., M.D.W., K.Y., A.H., H.I., P.D., T.K.K., M.H., V.K, A.G, A.B, S.S. and T.K. supported the ARPES experiment. T.T. and T.Sa. made and characterized single crystals of α- and β-Bi_{4}I_{4}, measured laser-microscope images, and performed transport experiments. T.Sh. performed the GISAXS experiment. R.N. measured SEM images. T.Sa., M.O. and R.A. calculated the band structure and analysed the band topology. R.N., K.K., M.D.W., T.Sa., M.O., T.Sh. and T.K. wrote the paper. All authors discussed the results and commented on the manuscript.

### Competing interests

The authors declare no competing interests.

### Corresponding authors

Correspondence to T. Sasagawa or Takeshi Kondo.

## Extended data figures and tables

### Extended Data Fig. 1 X-ray diffraction measurements for α- and β-Bi

_{4}I_{4}.**a**,**b**, X-ray diffraction measurements taken with CuKα radiation from the*ab*planes of α- and β-Bi_{4}I_{4}crystals used in our experiments. These two crystals were obtained by the quenching procedure (see Methods). The lattice parameters obtained by structural analysis using single-crystal X-ray diffraction are:*a*= 14.345 Å,*b*= 4.443 Å,*c*= 20.112 Å and*β*= 92.974° for the α-phase; and*a*= 14.406 Å,*b*= 4.432 Å,*c*= 10.475 Å and*β*= 107.463° for the β-phase.### Extended Data Fig. 2 Theoretical study of the electronic structure of α-Bi

_{4}I_{4}.**a**, The first Brillouin zone of α-Bi_{4}I_{4}.**b**, Bulk band structures calculated by GGA without SOC (left), with SOC (middle) and by GGA plus mBJ with SOC (right). The red and blue circles label the parities (±) for the bulk bands at the M and L points. The band topology of the α-phase is estimated to be a normal insulator (NI).**c**, Projected bulk bands for the top surface (001) of α-Bi_{4}I_{4}. The colour of the lines denotes different*k*_{z}values.### Extended Data Fig. 3 Theoretical study of the electronic structure of the (001) surface of β-Bi

_{4}I_{4}.**a**–**d**, Calculated bulk band dispersions for the \(\bar{{\rm{M}}}\) and \(\bar{{\rm{\Gamma }}}\) points of the (001) surface, depending on the topological phase. Colours denote different*k*_{z}values.**e**–**h**, Calculation of the band structures, including surface states (SS), for the \(\bar{{\rm{M}}}\) and \(\bar{{\rm{\Gamma }}}\) points of the (001) surface depending on the topological phase.### Extended Data Fig. 4 Spin texture of the quasi-1D Dirac state in β-Bi

_{4}I_{4}.**a**, Experimental geometry used for the spin-resolved measurements. We used*s*-polarization (vertical polarization) of incident light and measured three-dimensional spin polarizations,*S*_{x},*S*_{y}and*S*_{z}.**b**, Schematic illustration of the band structure for the spin-resolved measurements around \(\bar{{\rm{Z}}}\). The solid lines indicate the TSSs; the grey areas indicate the bulk valence band (BVB) and the bulk conduction band (BCB). The dotted lines denote the wavenumber (*k*_{y}) corresponding to the measured angles. The red and blue arrows show the direction of the measured spin polarization along*z*-axis.**c**, Spin texture of the constant energy contour for the (100) surface of β-Bi_{4}I_{4}obtained by first-principles calculations. The red arrows indicate the directions of the spin polarizations of the TSSs at each momentum point. The energy is set at 0.05 eV to duplicate the Fermi surface obtained in the ARPES experiment.**d**–**f**, Spin-resolved photoelectron intensities around \(\bar{{\rm{Z}}}\) (*θ*= 49°) for the*x*- (**d**),*y*- (**e**) and and*z*- (**f**) axes, where*x*is perpendicular to the (100) plane, while*y*and*z*are in-plane.**g**–**i**, Corresponding spin polarizations for the*x*- (**g**),*y*- (**h**) and*z*- (**i**) axes. Red and blue areas indicate positive and negative spin polarizations, respectively, of photoelectrons.### Extended Data Fig. 5 Angle-resolved photoemssion spectra of α- and β-Bi

_{4}I_{4}measured at*hv*= 85 eV and 6.994 eV.**a**, Experimental geometry of synchrotron-based ARPES at*hν*= 85 eV.**b**,**c**, Constant energy contours at*E*_{F}for α- and β-Bi_{4}I_{4}. The ARPES intensities are integrated within an energy window of 30 meV about the chemical potential. The dashed lines denote the two-dimensional Brillouin zone of the projected (001) surface.**d**,**e**, Band dispersion along the high-symmetry direction \(\bar{{\rm{\Gamma }}}\) – \(\bar{{\rm{M}}}^{\prime} \).**f**,**g**, Magnified intensity maps around the \(\bar{{\rm{\Gamma }}}\) point.**h**, MDCs at*E*_{F}around the \(\bar{{\rm{\Gamma }}}\) point.**i**, Energy-distribution curves (EDCs) at the \(\bar{{\rm{\Gamma }}}\) point.**j**,**k**, ARPES images measured at*hν*= 6.994 eV for the α-phase (**j**) and the β-phase (**k**) (these panels are duplicates of Fig. 3f, j (*θ*= 28°), respectively).**l**, EDCs along the red dotted lines in panels**j**,**k**. The black arrows indicate the peak positions of the bulk valence bands; the data clearly show the presence of bilayer splitting in the α-phase, consistent with our theoretical calculations (Extended Data Fig. 2c).### Extended Data Fig. 6 Polarization dependence of the quasi-1D TSS of β-Bi

_{4}I_{4}measured with a 6.994-eV laser.**a**, Experimental geometry of laser-ARPES, showing the direction of*s*- and*p*-polarized light.**b**, Fermi surface mapping obtained with*p*-polarized light.**c**,**d**, ARPES band maps across the \(\bar{{\rm{\Gamma }}}\) point of the (100) surface (red arrow in**b**) taken with*p*-polarized light (**c**) and*s*-polarized light (**d**).**e**,**f**, As for**c**,**d**, but with the momentum cut across the \(\bar{{\rm{Z}}}\) point of the (100) surface (yellow arrow in**b**). The quasi-1D TSSs are observed with both*p*- and*s*-polarized light, in agreement with the orbital characters obtained by first-principles calculations (Extended Data Fig. 7).### Extended Data Fig. 7 Orbital characters of the TSSs in β-Bi

_{4}I_{4}.**a**, Calculated constant energy contour for the (100) surface of β-Bi_{4}I_{4}. The energy is set at 0.05 eV to allow comparison with the Fermi surface obtained by ARPES experiment.**b**, Orbital separated spectra, calculated with the*p*-orbital model, of the surface electronic structure around the \(\bar{{\rm{\Gamma }}}\) point of the (100) surface (red arrow in**a**). Each subpanel shows the contribution of the*p*_{x},*p*_{y}and*p*_{z}orbitals for Bi, I and all elements. The directions of the*p*_{x,y,z}orbitals are defined relative to the (100) surface, as illustrated in**a**.**c**, The same results as**b**, but for the momentum cut across the \(\bar{{\rm{Z}}}\) point (orange arrow in**a**).### Extended Data Fig. 8 A slight modulation of 1D Fermi surface along

*k*_{z}in the side surface of β-Bi_{4}I_{4}.**a**, Calculated constant energy contour of the band for the (100) surface of β-Bi_{4}I_{4}. The energy is set at 0.05 eV to allow comparison with the Fermi surface obtained in the ARPES experiment.**b**–**e**, ARPES band maps and their MDCs at*E*_{F}around the \(\bar{{\rm{\Gamma }}}\) and \(\bar{{\rm{Z}}}\) points of the (100) surface. Double Lorentzian curves (black dotted curves) are fit to the MDCs with two-peak structure. The distances between these two peaks are indicated by the thin red arrows in**b**and**d**.### Extended Data Fig. 9 Periodicity of band dispersions obtained by surface-selective nano-ARPES at

*hv*= 74 eV.**a**,**b**, Schematics of surface preparation for nano-ARPES measurements of the side (**a**) and top (**b**) planes.**c**, Real-space mapping of photoemission intensity for the cleavage surface of**a**(left), along with a magnified image (right). The white circle indicates the position at which we performed surface-selective ARPES measurements.**d**, Fermi surface mapping for the (100) surface. The intensities are integrated within 100 meV. The dashed lines indicate the Brillouin zone for the (100) surface.**e**,**f**, Similar results to**c**,**d**, respectively, but for the (001) surface.**g**,**h**, ARPES band dispersion along the chain direction extracted at*k*_{y}= −0.4 Å^{−1}(blue arrow in**d**,**f**). The dashed red curves trace the band dispersions, with periodicities consistent with the sizes of the Brillouin zones for the (100) and (001) surfaces (red and black arrows in**g**and**h**, respectively). A high-flux nanometre-scale beam obtained using a Schwarzschild objective enabled clear mapping of*k*_{z/x}dispersions.**i**,**j**, ARPES band maps cut across the \(\bar{{\rm{\Gamma }}}\) and \(\bar{{\rm{Z}}}\) points of the (100) surface.**k**,**l**, ARPES band maps cut across the \(\bar{{\rm{\Gamma }}}\) and \(\bar{{\rm{M}}}\) points of the (001) surface.### Extended Data Fig. 10 Demonstration of the switchable topological state in Bi

_{4}I_{4}, using the same piece of crystal.**a**, Schematic showing the experimental sequence used to demonstrate the topological phase transition from the β- to the α-phase with the same piece of crystal.**b**,**c**, ARPES images of the same piece of crystal measured at*θ*= 0° and 20 K after quenching (**b**) and after mild cooling (**c**).**d**, Schematic showing the experimental sequence used to demonstrate the topological phase transition from the*α*- to the*β*-phase for the same piece of crystal.**e**,**f**, ARPES images of the same piece of crystal measured at*θ*= 0° and 20 K after mild cooling (**e**) and after quenching (**f**).

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