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1 In the format provided by the authors and unedited. DOI: /NPHYS4080 Topological states in engineered atomic lattices Robert Drost, Teemu Ojanen, Ari Harju, and Peter Liljeroth Department of Applied Physics, Aalto University School of Science, PO Box 15100, FI Aalto, Finland Present address: Varian Medical Systems Finland Oy, Paciuksenkatu 21, FI Helsinki, Finland NATURE PHYSICS 1

2 ADDITIONAL SPECTROSCOPY We also performed conductance spectroscopy at negative bias. No additional states are found up to -3 V bias on the vacancies (see Fig. S1). FIG. S1. Conductance spectroscopy at negative bias. Spectra measured on top of a Cl vacancy (black curve) and on the Cl-layer for reference (red curve). ALTERNATIVE DOMAIN WALL The topological nature of the band structure in the dimer chains demands that there be a domain wall state associated with any transition from one of the phases to the other. This is demonstrated in the manuscript with the example of a domain wall in which sequence of the alternating coupling constants is changed by replacing a strong bond with a weak one. The arrangement results in a sequence of three lattice sites coupled by weak bonds as shown in Fig. 2 of the manuscript. Another domain wall structure exists whereby a weak bond in the chain is replaced by a strong one. This results in a sequence of three lattice sites coupled by strong bonds. We constructed a dimer chain with such a domain wall an investigated its band structure to confirm the topological nature of the domain wall states. The chain is shown in Figure S2. The domain wall state is not localised to a single lattice site as it was the case with the structure shown in Fig. 2 of the manuscript. It instead appears on both of the outer sites of the domain wall trimer. As confirmed by our theoretical calculation of the LDOS, it is NATURE PHYSICS 2

3 FIG. S2. Alternative domain wall. a, Topography of the dimer chain with the alternative domain wall. b, Conductance map at 3.53 V (mid-gap energy). The domain wall state appears as an increased signal on the outer two sites of the vacancy trimer constituting the domain wall. c, Stacked contour plot of a series of conductance spectra acquired along the axis of the chain. The domain wall state is marked by the arrows. nevertheless a single electronic state which is spread across both lattice sites, thus fulfilling the topological requirement of a single mid-gap state per domain wall. ESTIMATING THE HOPPING PARAMETERS An quantitative calculation of the hopping parameters is made difficult as the exact energy of the vacancy states cannot be reliably extracted. A background subtraction using a reference spectrum acquired on the chlorine layer far away from any vacancies resulted in highly asymmetric line shapes and was abandoned as a result. To estimate the peak positions, we instead used the half maximum of the rising flank on the first peak as a marker. This approach can obviously only be used to estimate the position of the first resonance in each spectrum. The hopping parameters in the dimers can then be calculated NATURE PHYSICS 3

4 by subtracting the energy estimate for the bonding (lower energy) peak from that of a single vacancy. In order to estimate the next-nearest neighbour coupling in the Lieb lattice, we assumed an exponential dependence of the hopping parameters with distance. This model was fitted to the available data set, consisting of the long and short dimers shown in the manuscript text and a diagonal dimer with two vacancies touching each other at the corners. We stress that the reported coupling constants are estimates and should no be taken as quantitative results. EVENLY SPACED CHAINS We briefly review the properties of an evenly spaced chain as derived from tight binding calculations to show that the domain wall states we report in the main body of the manuscript cannot be understood as low energy states of the short segments of evenly spaced vacancies which constitute the domain walls. In the continuum limit, the Hamiltonian of such a chain is given by: 0 2t cos(k) H = (1) 2t cos(k) 0 The energy eigenvalues are found by solving det H ɛ 2 2 = ɛ 2 4t 2 cos 2 (k) =0 ɛ = ±2t cos(k) (2) The infinite chain would thus be metallic and would not support any localised states. The situation is of course different in finite systems where quantum confinement leads to the formation of discrete energy levels. Such a finite chain will support one zero energy mode if the number of sites is odd. The corresponding eigenstate will have constant amplitude on all odd sites and be zero on all even sites (Fig. S3a). A chain with even number of sites will not show any zero energy states. The states closest to the Fermi level have maximum amplitude at the edges of the chain and decay slowly towards the centre (Fig. S3b). NATURE PHYSICS 4

5 FIG. S3. Tight binding simulations on evenly spaced chain segments. a, Zero energy state of an evenly spaced chain with 31 sites (this state resides at the Fermi energy). Red asterisks denote the positions of the lattice sites. b, The state closest to the Fermi level of an evenly spaced chain with 30 sites. Red asterisks denote the positions of the lattice sites. LIEB LATTICE PRIMER The electronic band structure of the Lieb lattice can be derived from a tight binding treatment. The lattice is sketched in Figure S4. First, consider only nearest-neighbour coupling, that is, there are only connections from A-sites to B- and C-sites. Let the hopping parameter between adjacent sites be t, all other hoppings are zero. The lattice constant is set to 1 for convenience. The Hamiltonian for this system reads: 0 t(e ikx +e ikx ) t(e iky +e iky ) H = t(e ikx +e ikx ) 0 0 t(e iky +e iky ) 0 0 (3) In order to obtain the energy eigenvalues, one thus needs to solve: ɛ 2t cos(k x ) 2t cos(k y ) Det(H ɛ 3 3 )= 2t cos(k x ) ɛ 0 =0 (4) 2t cos(k y ) 0 ɛ Computing the determinant yields: ɛ[ɛ 2 +4t 2 cos 2 (k x )]+4ɛt 2 cos 2 (k y )=0 (5) By dividing both sides of (5) by ɛ, one finds the eigenvalues of the dispersing bands: NATURE PHYSICS 5

6 FIG. S4. Lieb lattice band structure. a, Sketch of the Lieb lattice with the assignment of the A, B, and C sites. b, Band structure in th first Brillouin zone with no next-nearest neighbour hopping. c, Band structure in the first Brillouin zone with t /t =0.3. ɛ = ±2t cos 2 (k x )+cos 2 (k y ) (6) In addition, equation (5) is always solved by setting ɛ to zero. This is the origin of the flat band. The full band structure is shown in Figure S4b. Now, add next-nearest neighbour couplings between the B- and C-sites. Let the nextnearest neighbour coupling constant be t. The Hamiltonian now becomes: 0 2t cos(k x ) 2t cos(k y ) H = 2t cos(k x ) 0 2t [cos(k x + k y ) + cos(k x k y )] 2t cos(k y ) 2t [cos(k x + k y ) + cos(k x k y )] 0 (7) Due to the new off-diagonal terms, the solution ɛ = 0 is no longer present. The most important influence of the next-nearest neighbour coupling is that the previously flat band acquires some dispersion the magnitude of which depends on t. The band becomes parabolic near the Brillouin zone centre, but remains nearly flat at the corners. For an illustration, see Figure S4c. Consider now the eigenvectors corresponding to the eigenvalues in the case with only nearest neighbour coupling. The eigenvector for the ɛ = 0 eigenstate is a non-trivial solution to the equation NATURE PHYSICS 6

7 0 2t cos(k x ) 2t cos(k y ) v 1 2t cos(k x ) 0 0 v 2 =0 (8) 2t cos(k y ) 0 0 v 3 This leads to a set of equations 2t cos(k x )v 2 2t cos(k y )v 3 =0 (9) 2t cos(k x )v 1 =0 (10) 2t cos(k y )v 1 =0 (11) Equations (10) and (11) are only solved for v 1 = 0 while (9) implies that v 2 = cos(k y ) v 3 = cos(k x ) (12) For the non-zero eigenvalues, the eigenvector is found by solving ±2t cos 2 (k x )+cos 2 (k y ) 2t cos(k x ) 2t cos(k y ) 2t cos(k x ) ±2t v 1 cos 2 (k x )+cos 2 (k y ) 0 2t cos(k y ) 0 ±2t v 2 =0 cos 2 (k x )+cos 2 (k y ) v 3 (13) leading to the system ±2t cos 2 (k x )+cos 2 (k y )v 1 2t cos(k x )v 2 2t cos(k y )v 3 =0 (14) 2t cos(k x )v 1 ± 2t cos 2 (k x )+cos 2 (k y )v 2 =0 (15) 2t cos(k y )v 1 ± 2t cos 2 (k x )+cos 2 (k y )v 3 =0 (16) As the right hand side of equation (13) is zero, one of the entries of the vector v may be set to 1 without loss of generality. Choosing v 3 = 1 and plugging into (15) and (16) yields v 1 = cos 2 (k x )+cos 2 (k y ) cos(k y ) v 2 = cos(k x) cos(k y ) (17) NATURE PHYSICS 7

8 The full normalized eigenvectors are thus v 0 = 1 0 cos(k a y ) cos(k x ) v ± = 1 2a cos 2 (k x )+cos 2 (k y ) cos(k x ), (18) cos(k y ) where a = cos(k x ) 2 + cos(k y ) 2. From Eq. 18, one can see that the zero energy eigenstate is derived from them B and C sites of the lattice only while all three sites contribute to the dispersive eigenstates with ɛ 0. To see that B and C sites contribute equally to the flat band state, we transform the sublattice amplitudes back to real space: Ψ B 0 ( r B )= 1 N e i k r cos(k B y) a k, Ψ C 0 ( r C )= 1 N e i k r cos(k C x) a k (19) Here r B/C denote the positions of B and C lattice sites and N is the number of unit cells in the system. Fixing the origin (0, 0) to an arbitrary A lattice site, the B and C sites in the same unit cell have coordinates r B = (0, 1), r C = (1, 0). Since the possible values for k x and k y are the same, we see that the sublattice amplitudes in the same unit cell satisfy Ψ B 0 ((0, 1)) = Ψ C 0 ((1, 0)). Thus the sublattice probabilities of the flat band, obtained by squaring the amplitudes, are equal. This conclusion holds for an arbitrary unit cell in the lattice. TIGHT BINDING SIMULATIONS OF THE LIEB LATTICE Based on our estimates of the coupling constants outlined above, we performed tight binding calculations on the Lieb lattice, as mentioned in the manuscript. Despite the rough estimate of the coupling constants, simulations with a nearest neighbour coupling of t NN 0.14 ev and a ratio of next-nearest neighbour to nearest neighbour coupling of r NNN = t NNN /t NN 0.33 reproduce the experimental results far better than those with higher or lower ratios r NNN, see Fig. S5 for an illustration. The on-site energy was fixed to 3.5 V. The LDOS spectra and maps were calculated with an estimated broadening of 0.17 ev for all states. The pronounced shoulder at 3.1 V in the spectra measured on the A lattice site is best reproduced for the simulated spectrum with r NNN =0.33. It is absent at lower ratios and splits into a double shoulder for higher r NNN, which is not observed in the experiment. NATURE PHYSICS 8

9 FIG. S5. Tight binding simulations with different coupling ratios. a, LDOS on the A lattice site. An experimental curve is shown in red for reference. The simulated curves use the ratios r NNN = 0 (blue), 0.34 (purple), and 0.5 (green). Curves are offset for clarity. b - e, Simulated conductance maps at 2.85 V, 3.15 V and 3.5 V (same as the experimental maps in the manuscript) with r NNN = 0 (top) and r NNN = 0.33 (bottom) Some key features of the experimental conductance maps are also better reproduced using r NNN =0.33. In particular, the states are more confined towards he centre of the lattice at E =2.85 V, but more uniformly distributed at E =3.15 V. At the nearly flat band, simulations with r NNN =0.33 reproduce the increased signal at the centre of the edges of the structure. In addition, the signal at the corners of the structure is also increased. We speculate that the strong signal on the corner sites observed in the experiment is due to coupling to states in the conduction band of the chlorine layer. LIEB LATTICE MAPS ABOVE THE CL LAYER BAND EDGE We also acquired conductance maps above the single vacancy level of 3.5 V (see Fig. S6). At these energies, the conduction band of the chlorine layer contributes significantly to the signal. It is difficult to quantitatively substract this contribution from the states of the Lieb lattice. NATURE PHYSICS 9

10 FIG. S6. Conductance map above the Cl layer band edge. a, Topography of the Lieb lattice. b, Conductance map acquired at 3.6 V, ca 100 mv above the band edge of the Cl layer. The conduction band of the chlorine layer dominates the contrast and overshadows the states of the Lieb lattice near the edges of the structure. NATURE PHYSICS 10