Abstract
A quantum state of matter that is forbidden to interact with photons and is therefore undetectable by spectroscopic means is called a dark state. This basic concept can be applied to condensed matter where it suggests that a whole band of quantum states could be undetectable across a full Brillouin zone. Here we report the discovery of such condensed-matter dark states in palladium diselenide as a model system that has two pairs of sublattices in the primitive cell. By using angle-resolved photoemission spectroscopy, we find valence bands that are practically unobservable over the whole Brillouin zone at any photon energy, polarization and scattering plane. Our model shows that two pairs of sublattices located at half-translation positions and related by multiple glide-mirror symmetries make their relative quantum phases polarized into only four kinds, three of which become dark due to double destructive interference. This mechanism is generic to other systems with two pairs of sublattices, and we show how the phenomena observed in cuprates, lead halide perovskites and density wave systems can be resolved by the mechanism of dark states. Our results suggest that the sublattice degree of freedom, which has been overlooked so far, should be considered in the study of correlated phenomena and optoelectronic characteristics.
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Acknowledgements
This work was supported by the National Research Foundation (NRF) of Korea funded by the Ministry of Science and ICT (grant nos. NRF-2021R1A3B1077156, NRF-RS-2024-00416976 and NRF-RS-2022-00143178 to K.S.K.; NRF-2020R1I1A3073680 to K.L.), Yonsei Signature Research Cluster Program (2023-22-0004) to K.S.K. and Industry-Academy Joint Research Program between Samsung Electronics and Yonsei University to Y.Y. This research used resources of the Advanced Light Source, which is the Department of Energy, Office of Science User Facility, under contract no. DE-AC02-05CH11231. We acknowledge the Diamond Light Source for time on Beamline I05 under proposals SI30270 and SI35764.
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M.K. performed the ARPES experiments on PdSe2 with help from Y.C., C.J., E.R. and A.B. Y.C. and Y.K. performed the ARPES experiments on cuprates with help from T.K.K., C.C., C.J., E.R. and A.B. Y.K. performed the ARPES experiments on CsPbBr3 with help from S.C., Y.C., T.K.K., C.C., J.D., C.J., E.R. and A.B. J.W.P., J.P., Y.Y., D.S., J.H.R. and K.L. synthesized and provided the samples. Y.C. carried out the tight-binding band calculations and spectral simulations. K.S.K. conceived and supervised the project. Y.C., M.K., Y.K. and K.S.K wrote the manuscript with contributions from all other co-authors.
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Extended data
Extended Data Fig. 1 ARPES data of PdSe2 in kx and ky.
a-p, Complete set of ARPES data taken at 80 K along kx (a-h) and ky (i-p). The light polarisation and scattering plane are shown by p-pol / xz (a,e,i,m), s-pol / xz (b,f,j,n), p-pol / yz (c,g,k,o), and s-pol / yz (d,h,l,p), as marked on the top. The photon energy is 87 eV for kz = Γ6 (a-d,i-l) and 119 eV for kz = Γ7 (e-h,m-p), as marked on the left. After background normalisations, the same colour scale is applied to each pair of data taken by p-polarised light and s-polarised light. Grey dotted lines indicate the expected location of valence bands. It can be identified that the valence bands of PdSe2 centred at all Γ points but Γ006 are dark states, regardless of scattering planes.
Extended Data Fig. 2 ARPES data of PdSe2 in kz and ks.
a-d, Complete set of ARPES data taken at 80 K along kz. The light polarisation and scattering plane are shown by p-pol / xz (a), s-pol / xz (b), p-pol / yz (c), and s-pol / yz (d), as marked on the top. e-l, Complete set of ARPES data taken along ks. The light polarisation and scattering plane are shown by p-pol / s1z (e,i), s-pol / s1z (f,j), p-pol / s2z (g,k), and s-pol / s2z (h,l), where s1 and s2 are two diagonal directions, as shown on the top. The photon energy is 87 eV for kz = Γ6 (e-h) and 119 eV for kz = Γ7 (i-l), as shown on the left. After background normalisations, the same colour scale is applied to each pair of data taken with p-polarised and s-polarised light. Grey dotted lines indicate the expected location of valence bands. It could be identified that the valence bands of PdSe2 centred at Γ007 and Γ009 are dark states, regardless of scattering planes.
Extended Data Fig. 3 ARPES data of PdSe2 in constant energy maps.
a-p, ARPES data taken at 80 K and plotted at E = –0.4 eV as a function of kx and ky. The light polarisation and scattering plane are indicated by p-pol / xz (a,e,i,m), s-pol / xz (b,f,j,n), p-pol / yz (c,g,k,o), and s-pol / yz (d,h,l,p), as marked on the top. The photon energy is 87 eV for kz = Γ6 (a-d), 119 eV for kz = Γ7 (e-h), 154 eV for kz = Γ8 (i-l), and 197 eV for kz = Γ9 (m-p), as shown on the left. After background normalisations, the same colour scale is applied to each pair of data taken with p-polarised light and s-polarised light. Grey dotted lines show zone boundaries. We found that the valence bands of PdSe2 in 13 out of the 18 Brillouin zones are dark states, regardless of scattering planes. The remnant intensity seen in l,n are due to Se contributions, as more fully discussed in Extended Data Fig. 7.
Extended Data Fig. 4 Photon energy dependence of dark states.
a, Alternating relative phases of PdSe2 between kz = Γ2n and Γ2n+1. b,c, Photon-energy dependence of ARPES data taken at kxy = Γ10 (b) and Γ01 (c), as indicated by dotted arrows in a (p-pol + s-pol, xz scattering plane). The colour scale is normalised based on the background with respect to those taken at kxy = Γ00 (Extended Data Fig. 2a, b), as shown in d. d, Momentum distribution curves taken at E = −0.1 eV along kz (80−200 eV) to make a comparison for those at Γ10, Γ01, and Γ00. e-g, Line profiles taken at 4 different Γhkl points collected to separately show the photon-energy dependence of red (e), yellow (f), and green (g) states corresponding to dark states. As seen in d-g, we find little photon-energy dependence of dark states for all photon energies used here. This is because multiple Coulomb wavelets become simplified (or polarised) into only four kinds in materials with two pairs of sublattices connected by glide-mirror symmetries. Then, the selection rule becomes as stringent as in atomic systems since the role of orbital angular momentum properties is replaced by the equally well-defined sublattice degree of freedom. Namely, this is a consequence of such fully polarised initial state wavefunctions, which has nothing to do with final states, and hence no dependence on photon energies.
Extended Data Fig. 5 ARPES data of PdSe2 compared with simulations.
a-p, ARPES data and simulations of band dispersion in kx (a-h) and constant energy maps at E = –0.4 eV (i-p). The ARPES data or simulations and light polarisation are indicated by Data / p-pol (a,e,i,m), Data / s-pol (b,f,j,n), Sim. / p-pol (c,g,k,o), and Sim. / s-pol (d,h,l,p), as marked on the top. The photon energy is 87 eV for kz = Γ6 (a,b,i,j) and 119 eV for kz = Γ7 (e,f,m,n), as marked on the left. After background normalisations, the same colour scale is applied to each pair of data taken by p-polarised light and s-polarised light. Grey dotted lines show the expected location of valence bands in a-h and zone boundaries in i-p. The bright and dark states can be reproduced by spectral simulations based on the model of sublattice interference.
Extended Data Fig. 6 Doping and temperature dependence of Fermi arcs.
a-d, Fermi surfaces of single-layer cuprates simulated with our model (Methods) at U = 0.4t1 by shifting EF for the hole concentration (p) marked on top of each panel. e-h, Fermi surfaces of single-layer cuprate simulated with our model at p = 0.17 by varying U/t1 as marked on top of each panel. Our model of sublattice interference reproduces key aspects of phenomenology in doping and temperature dependence of Fermi arcs17,19: Those in a-d reproduce a key feature in doping dependence that the length of Fermi arcs grows in size with p. More importantly, the anti-nodal gap reduces in magnitude with increasing either doping or temperature. This can also be well reproduced by our simulations in e-h by reducing U/t1 that represents the strength of symmetry breaking, no matter whether its origin is CDW or antiferromagnetism.
Extended Data Fig. 7 Se contributions in the ARPES data of PdSe2.
a,b, Crystal structure of PdSe2 illustrated by the ball-and-stick model and viewed from the top (a) and the side (b). The 8 Se atoms in the primitive cell are labelled A1–D1 and A2–D2. c,d, ARPES data of PdSe2, taken by the s-polarised light of 87 eV (kz = Γ6) at the scattering plane of xz (c) and yz (d), and plotted at E = –0.4 eV as a function of kx and ky. e,f, Part of constant-energy maps marked in c,d by red squares and shown in the extremely narrow colour scale. g,h, ARPES data of PdSe2 shown in ky (g) and kx (h) along the grey dotted lines in e,f, respectively. i,j, Relative phases of ϕA1C1/B1D1 (i) and ϕA1B1/C1D1 (j) obtained from tight-binding models and shown at kz = Γ2n for the same area as in e,f (ϕA1C1/B1D1 = ϕC2A2/D2B2, ϕA1B1/C1D1 = ϕA2B2/C2D2). The colour scale in units of radian is given at the bottom. k-n, Corresponding simulations of constant-energy maps (k,l) and band dispersions (m,n) with relative phases in i,j and sublattice interference. There is overall little ARPES intensity on the valence band of PdSe2 taken by s-polarised light in c,d. However, if narrowing down the intensity scale for the area marked by the red square, we found there is a remnant feature with the intensity pattern that vanishes at ky = 0 along the kx axis in e and at kx = 0 along the ky axis in f. Even these characteristic intensity patterns can be reproduced well by ARPES simulations based on the model of quantum interference between 8 Se atoms (see Methods), as shown in k-n.
Extended Data Fig. 8 ARPES simulation for charge density waves.
a,b, Crystal structure of monoatomic chains running along x in normal (a) and CDW (b) states. The two sublattices formed by dimerization are labelled A and B in b. c,d, Band structure of monoatomic chains in normal (c) and CDW (d) states calculated by tight-binding models and plotted along kx. e, ϕAB of the lower energy band in d obtained from tight-binding models and shown along kx. f,g, ARPES simulations for normal (f) and CDW (g) states based on the model of sublattice interference. With the scattering plane of xz, Mk can be written in the form of \(1+{e}^{i{\phi }_{\text{AB}}}\) arising from integration over the scattering plane. This is independent of light polarisation because no parity conversion is expected across this scattering plane. The Fermi energy is assumed to be at the grey horizontal dotted lines. h, Magnified view of ARPES simulations marked by the red box in g, which shows typical spectral features that gradually diminish in the vicinity of energy gaps or along backfolding of the main band. This can be explained by incomplete phase polarisations near ±π/2a indicated by red arrows in e as a natural consequence of the two sublattices located slightly off the half-translation positions in b. Those centred at ±π/a with ϕAB ≈ ±π cannot be detected by ARPES, because they are nearly the dark states.
Source data
Source Data Fig. 1
Source data for Fig. 1h.
Source Data Fig. 2
Source data for Fig. 2a,d,g.
Source Data Fig. 5
Source data for Fig. 5c,d.
Source Data Extended Data Fig. 4
Source data for Extended Data Fig. 4d–g.
Source Data Extended Data Fig. 8
Source data for Extended Data Fig. 8e.
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Chung, Y., Kim, M., Kim, Y. et al. Dark states of electrons in a quantum system with two pairs of sublattices. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02586-x
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DOI: https://doi.org/10.1038/s41567-024-02586-x
