Uncertainty principle for experimental measurements: Fast versus slow probes

The result of a physical measurement depends on the time scale of the experimental probe. In solid-state systems, this simple quantum mechanical principle has far-reaching consequences: the interplay of several degrees of freedom close to charge, spin or orbital instabilities combined with the disparity of the time scales associated to their fluctuations can lead to seemingly contradictory experimental findings. A particularly striking example is provided by systems of adatoms adsorbed on semiconductor surfaces where different experiments – angle-resolved photoemission, scanning tunneling microscopy and core-level spectroscopy – suggest different ordering phenomena. Using most recent first principles many-body techniques, we resolve this puzzle by invoking the time scales of fluctuations when approaching the different instabilities. These findings suggest a re-interpretation of ordering phenomena and their fluctuations in a wide class of solid-state systems ranging from organic materials to high-temperature superconducting cuprates.

Scientific RepoRts | 6:19728 | DOI: 10.1038/srep19728 configuration (i.e. valence states). A similar contradiction, also pointing towards the importance of charge-degrees of freedom, is found between ARPES and low-energy electron diffraction (LEED) and has led to speculations about inhomogenous phases 45 , with thermally fluctuating Sn-positions 16,17,26,[46][47][48] . On the other hand it has been shown 43 that Sn/Si(111) is in the immediate vicinity of a phase transition between a Mott insulating and a charge-ordered phase. Here we show that this peculiar position is key to resolve the above puzzles: Close to the phase transition we find dynamic charge fluctuations (beyond static approximations like LDA) of 3 × 3 and ( × ) 2 3 2 3 R30° symmetry with lifetimes on the order of femto seconds. This lifefime is long enough for probes like ARPES and core level photoemission (cPES) to detect them while STM, as a static (i.e. time averaging) probe, is blind to such dynamic modes. Our analysis becomes quantitative when assuming a (likely) phase separation in the vicinity of the first order Mott-CO transition.

Results
Our starting point is the charge-charge correlation function computed within combined many-body perturbation theory and dynamical mean field theory (GW + DMFT), see upper left-hand side of Fig. 1, ref. 43, and the Methods section. This quantity suggests the presence of three major dynamic charge-fluctuation symmetries, which are represented in a cartoon-like fashion in the upper right-hand side of Fig. 1): a 3 × 3 charge-ordered state, where three inequivalent sites are respectively doubly occupied, half-filled and empty (a "210" charge distribution), a ( × ) 2 3 2 3 R30° reconstructed state where stripes of empty sites alternate with stripes of doubly occupied sites, and the conventional Mott insulating state where all sites are half-filled. Its time/frequency dependence indicates fluctuations of these symmetries with a characteristic timescale (i.e. quasiparticles of a specific lifetime) of the order of femto seconds, long enough for the cPES process to capture the fluctuations in a "snapshot"-like measurement. We remark that for 3 × 3 charge-ordered Sn/Ge(111) a charge distribution with one filled and two quarter-filled sites per unit cell has been discussed. The configurations of the many-body wave functions that would lead to such charge distribution in the symmetry broken phase are included in our GW + DMFT calculations as charge fluctuations visible in our calculated charge susceptibility.
Mathematically, we determine the ground state wave function to be a quantum superposition state (QS) composed of the three configurations with weights 0.13, 0.56 and 0.31 for the ( × ) 3 3 R30°, 3 × 3, and ( × ) 2 3 2 3 R30° configurations respectively, see Methods section. With these insights, we now turn to a discussion of the different observables measured in cPES, STM and ARPES. It is the sum of the weighted contributions of the two coexisting phases close to the Mott-CO insulator transition (blue and orange dashed lines). The solid narrow lines (narrow peaks) resolve the contributions to the total spectrum by empty surface orbitals (red), singly occupied surface orbitals (blue) and fully occupied surface orbitals (green) incorporating respective multiplet splittings. Core-level spectroscopy. The first probe we consider is core electron emission from the Sn 4d-shell.
Core-level spectroscopy is a local Sn probe as sketched on the left-hand side of Fig. 1 where a 4d core electron is emitted out of the solid by an incoming photon. Due to the Coulomb interaction of this 4d core hole with the 5p valence electrons (U cv ) the spectrum is sensitive to the Sn valence configuration which consists of an either empty, half-filled, or full surface orbital. It is therefore a most efficient probe for charge-ordered states or charge fluctuations on timescales slower than the experimental process. To arrive at a first principles description of the core-level spectra we first determine the core-level emission spectra corresponding to the three different valence configurations (blue, red and green curves in the central panel of Fig. 1 corresponding to singly occupied, empty and full surface orbital configurations respectively) from cluster multiplet simulations (see Methods section). The main energy scale for each contribution is the spin-orbit coupling of the core hole which splits each spectrum into two main peaks associated to a core hole with total angular momentum J ch = 5/2 or J ch = 3/2. On a smaller energy scale (below experimental resolution), the core-valence interaction (U cv ) leads to multiplet splittings within each J ch subspace (Note that for a filled valence shell (green) and its fully spherical charge distribution such multiplet splittings cannot occur).
If the system were homogeneously in the QS state determined above the resulting core-level spectra would be given by the superposition of the spectra determined for the different valence states with contributions of ≈ 13% of the (  ×  )  3 3 R30° phase, ≈ 56% of the 3 × 3 phase, and ≈ 31% of the ( × ) 2 3 2 3 R30° phase. Translated into Sn valence contributions this corresponds to ≈ 32% half-filled and ≈ 68% empty/doubly-occupied sites. The resulting spectrum is plotted as yellow-dashed line in Fig. 1), and is found to give an unsatisfactory description of the experimental data (black 45 /gray 22 dots). If on the other hand, the system were in a pure Mott insulating state, our estimate of the core-level spectrum would be given by the contribution of the half-filled surface orbital only, broadened by the experimental resolution (blue dashed line in Fig. 1). Obviously, this assumption does not hold either, confirming our analysis of charge fluctuations contributing to the core-level spectrum. Being an order to order transition, the Mott to CO transition is expected to be of first order. This is confirmed by the behaviour of our charge susceptibility (discussed further below). Phase separation is very likely in the vicinity of such a first-order transition.
Indeed, closer inspection of the spectra corresponding to the two possible states and comparison to the experimental spectra shows that while neither the homogeneous QS state of the type determined above nor the Mott state yield theoretical spectra in agreement with experiment, the sum of the two spectra with weights 0.7 for the QS state and 0.3 for the Mott state, does. Such an incoherent superposition can be interpreted as simulating a spatial averaging of the two phases, that is, a state where phase separation leads to a spatial coexistence of Mott-insulating and charge-ordered patches with ratio 3/7. The obviously good agreement with the experimental measurement gives support to our interpretation of the sample being in an inhomogeneous state where Mott insulating islands are embedded into a dynamical QS background, and the spatial averaging done by the core-level spectroscopy results in a weighted average of the two contributions with weights 3:7. These results yield a complementary, quantum mechanical perspective on speculations of an inhomogeneous phase 45 , although rather as a superposition of the Mott state with the QS precursor we have described, locating Sn/Si(111) in the phase coexistence region of a first-order phase transition between these two phases.
Comparison to the related compound Sn/Ge(111) 26,46,47 yields further insight: experiments unambigously find Sn/Ge(111) in a fully static charge-ordered phase of 3 × 3 symmetry. In this case the phase coexistence has disappeared and the QS patches have grown to macroscopic length scales at temperatures below 60 K. Moreover, for the static case the experimental timescale is irrelevant and direct comparison of cPES and STM is unproblematic. In Sn/Si(111), on the other hand, such comparison can only be made by considering the snapshot-like nature of cPES measuring the spatial average of coexisting phases. Before turning to a quantitative discussion of the time and spatial extent of the charge fluctuations, based on the charge-charge correlation function χ GW+DMFT as obtained from GW + DMFT (see methods section) we revisit another experiment -complementary to STM and cPES -of Sn/Si(111) that has caused recent controversies.
Angle-resolved photoemission spectroscopy. Just as in core-level spectroscopy, the comparison of ARPES with STM surface images (suggesting the absence of any charge order) is not straightforward. More specifically, previous interpretations of ARPES spectra have assumed the breaking of some kind of spin-27 or charge-44 symmetry of the ground state in order to account for backfolded features of the momentum-resolved spectral function. The origin of such a symmetry breaking has however been unclear so far, since there is no other experimental evidence for it: a static charge-ordered state would contradict STM results, and no experimental probe has found any direct indication for spin-ordering. In the light of the previous discussion on the core-level spectroscopy and by comparison of experimental ARPES spectra with theoretical calculations, we will argue in the following that assuming such symmetry breaking is unnecessary when taking into account the typical timescale of the experiment: just like core-level spectroscopy, ARPES can be understood as a "snapshot" probe that spatially averages over the surface. Hence, short-lived charge fluctuations of specific symmetry and finite but sufficient spatial extension (see next paragraph for details and quantification) will be picked up by ARPES and incoherently averaged (such an effect has been recently shown for AF spin fluctuations and their impact on ARPES experiments for the high T c cuprates 49 ).
In order to provide a direct comparison between our theory and experiment we have simulated the ARPES spectrum in the (  ×  )  3 3 R30°-, the 3 × 3-, and the ( × ) 2 3 2 3 R30°-phase. In Fig. 2 we show in the bottom right panel the experimentally obtained ARPES signal along a specific path in the ( × ) 3 3 R30° Brillouin zone (BZ) (taken from ref. 27). Along with the experimental data, we also provide a complete map of single-particle spectra, so as to illustrate how the total spectrum can be disentangled into its components and provide reference Scientific RepoRts | 6:19728 | DOI: 10.1038/srep19728 spectra for future experiments on Sn/Si(111) or related compounds in the X/Y(111) family (X being a group IV adatom and Y a semiconductor like Si, Ge, SiC, etc.).
If we now take the mixture (and subsequent broadening) of the three theoretically calculated spectra (left panels in in Fig. 2) with the weights determined above, the agreement between experimental spectra and theory becomes satisfactory. To be precise, we can identify certain symmetry features, i.e. backfoldings, to be related to a specific charge fluctuation. The common feature of all shown spectra is their insulating nature, i.e. a finite gap. In the (  ×  )  3 3 R30°-phase (upper left) the gap separates an upper and lower Hubbard band of the Mott insulating state In the 3 × 3 phase (center left), we find (as expected from the 3 × 3 occupations) the combination of a band-insulating (empty and doubly occupied sites) and Mott-insulating (singly occupied sites) gap. Finally, the spectral function of the ( × ) 2 3 2 3 R30°-stripe phase (lower left) separates the bands of the doubly occupied and empty lattice sites and, hence, represents a band-insulating spectrum. While the momentum-structure of the (  ×  )  3 3 R30° Hubbard bands closely resembles the dispersion of a free electron on the surface lattice, the CO phases display characteristic backfolding features in the ( × ) 3 3 R30° BZ. Particularly noteworthy are maxima of the spectral functions along the K − Γ and M − Γ directions (attention must be paid to the different conventions for the naming of BZ points in different publications) that were subject to discussions in previous studies 27, 44 . Finally, we remark that the difference of the total energy scale (i.e. the Fermi level) between our simulation and experiment (Δ ε F ~ 0.15-0.2 eV) most probably originates from i) uncertainties of the theoretical/ experimental determination of the Fermi level or ii) error bars of our ab initio calculated values for onsite and non-local interactions.
Time and spatial resolution of the charge fluctuations. In the previous two paragraphs we have seen that spectroscopies such as core electron emission and ARPES seem to suggest charge order and, hence, are in contradiction to STM images of the Sn/Si(111) surface. As alluded to before, this contradiction can be resolved by considering the typical timescales of the experiments: While the spectroscopies are spatially averaged but quasi instantaneous snapshot probes, the STM complementarily yields time-averaged but spatially resolved information. We will now report on the details of time and spatial resolution of the charge fluctuations relevant to the phase coexistence. Indeed, the discussion on the different experiments above and their interpretation is based on a result from our theoretical ab-initio treatment of Sn/Si(111) within self-consistent GW + DMFT applied to a low energy Hamiltonian (see Methods section). The most relevant quantity for the present discussion is the charge-charge correlation function χ (q, ω) resolved in momentum q and frequency ω and its respective Fourier transforms. In our framework this quantity is self-consistently obtained and can be employed as a sensitive probe  Fig. 1). The red dashed line marks the Fermi energy (ε F = 0). Right hand side panels: Weighted sum of A(k, ω) of the contributions shown on the left hand side (top). Electron removal part of the total spectral function with additional broadening (middle) for comparison with experimental ARPES data (bottom). Note that our simulation has no information about k-dependent matrix elements of the actual ARPES measurement so that relative intensities of theory and experiment are not expected to be comparable.
for charge-order instabilities. More specifically, the vicinity to a transition into an ordered phase of a specific symmetry would be signaled by the behaviour of χ (q, ω = 0) at the corresponding q vector. Intuitive insight can be obtained from the Fourier transformed correlation function in real space and time χ (R, τ) plotted in Fig. 3. With this quantity we can find the typical correlation length ξ and timescale Δ τ 0 of a charge fluctuation. Since the Mott to CO phase transition is not a second order transition, χ (R, τ) does not become continuously long range (i.e. χ (q, ω = 0) does not diverge at a given q) but the correlation length ξ → ∞ only increases up to a finite value of the order of a few lattice constants before entering the symmetry-broken phase.
In Fig. 3 we plot the calculated χ GW+DMFT (R, τ) for Sn/Si(111) on the z-axis at three different imaginary time slices which correspond to averages over increasingly large timescales. Four panels show the evolution from instantaneous measurements to averages roughly over some femto seconds. x-and y-axes present the 2D surface indicated also by the black dots at the respective adatom sites. From these plots we can conclude that the Mott phase coexists with short-lived finite size lattice-commensurate charge fluctuations. To quantify this claim we extract a correlation length at τ = 0 of about 4 l.u. (enough for backfolded bands to be occupied in the ARPES experiment 49 ). These numbers immediately resolve the spectroscopy vs. microscopy puzzle: Sn/Si(111) is found in close vicinity but not yet in a charge-ordered phase (which can actually be reached by substituting the Si substrate by a Ge one). However, the ordered phase is preceded by quickly decaying charge fluctuations that can be picked up by fast core-level and photoemission spectroscopies but not by STM.

Discussion
In this work, we have demonstrated that the apparent contradictions between STM, ARPES and core-level spectroscopy for two-dimensional systems of adatoms adsorbed on semiconductor surfaces can be resolved by considering that i) specific compounds like Sn/Si(111) are located in the phase coexistence region of the first-order phase transition from a Mott insulator to a charge-ordered insulator, and ii) the timescales intrinsic to the different experiments matter: quickly decaying charge fluctuations (of specific symmetries) can be seen by fast snapshot-like spectroscopies (core-level spectroscopy, ARPES) while slow microscopy (STM) detects only a time-averaged image in which the charge modulations are averaged out. We have shed light on the history of controversial interpretations of Sn/Si(111) by quantifying these statements, based on first principles many-body calculations using combined many-body perturbation theory and extended dynamical mean-field theory (GW + DMFT). In order to provide a direct theory-experiment comparison, we have computed the observables of core-level spectroscopy and angular resolved photoemission. Moreover, we have visualized and discussed the key observable for dynamically fluctuating surface compounds: the charge susceptibility. Our analysis underlines the need for a very careful analysis of experimental results in circumstances where characteristic timescales of the material (i.e. fluctuations) and the experimental probe coincide.
Interestingly, in the related Sn/Ge(111) system, experimental discrepancies at temperatures above the 3 × 3 ordered phase have been explained with single particle theories by a freezing-in of Sn-vibrations along the surface normal 46 while no such freezing-in is seen in Sn/Si(111). It has been speculated that physics equivalent to Sn/ Ge(111) might occur only at lower temperatures 45 which, however, has not been confirmed so far. Other paradigmatic phase transitions have also been explained by dynamical fluctuations, as in the metal-insulator transition on In/Si(111) 50-52 or in the novel cluster-diffusion transition on Sn/Si(111):B 53 . In all of these transitions, the fluctuations have been argued to correspond to classical fluctuations between inequivalent configurations. While we do not exclude backcoupling to the lattice, our study reveals that the experimental observations can be explained from the purely electronic many-body wavefunction.
It is clear that this kind of phenomenology is not restricted to adatom systems, but can be expected to occur quite generally in two-dimensional systems close to competing instabilities. Most notably, observations of charge-ordering fluctuations dominate the recent literature on high-temperature superconducting oxides 54-57 , with conflicting interpretations concerning stripe-or checkerboard-type charge-ordering tendencies, their driving mechanism and their implications for superconductivity. These questions should be carefully reexamined in the light of our findings. The present surface systems only provide a particularly clean and tunable model system, without the complications due to disorder, mixing in of multi-orbital or ligand degrees of freedom present in the cuprates. The well-defined single-orbital character of the surface systems allows for a truly first principles treatment, providing us with a unique tool for refining our understanding of competing instabilities in the proximity of different ordering phenomena. Our results and conclusions do not negate magnetic fluctuations (discussed by Li et al. ref. 27 and a very recent study by Glass et al. ref. 64) which should be considered as complementary to the discussed charge fluctuations. Reconciling both spin-and charge degrees of freedom in one theoretical framework will be one of the next challenges -fortunately the materials to be tested are real and experimental data is accessible to support or falsify theoretical predictions.

Methods
Charge correlation function. The correlation function in the charge channel χ (R, τ) displayed in Fig. 3 has been obtained by spatial and temporal Fourier transformation of the charge correlation function χ (k, iω). The latter is computed from the polarization function P (k, iω) through the relation: Here, v (k) is the Fourier transform of the interactions ( ) (a in the lattice constant and R i denotes a lattice site.). The polarization function is computed in the GW + DMFT approximation [58][59][60][61] , namely as the sum of the impurity polarization P imp (iω) and of the nonlocal part of the bubble ~2 GG, more specifically where G (q, iv) is the fully self-consistent Green's function from a converged GW + DMFT calculation.
The factor of 2 stems from spin degeneracy. The values of the interaction parameters are calculated within the constrained random phase approximation 42,43 , namely U = 1.0 eV and V = 0.5 eV. A recent ab initio determination of the interaction parameters of the Si α-phases (X/Si(111) with X = C, Sn, and Pb) has found nonlocal interactions to be as large as 50% of the onsite ones and established a materials trend with the Sn compound being "half-way" between Mott insulating C/Si(111) and charge-ordered Pb/Si(111) 62 .
Ground-state wave function of charge-ordered state. In order to determine the weight with which the "210", stripe and Mott configurations contribute to the ground-state wave function in the charge-ordered state, we have solved -by exact diagonalization -a six-site cluster with periodic boundary conditions. Subsequent projection of the ground state on the three relevant configurations of interest results in the estimates for the coefficients shown in Fig. 4 as a function of non-local interaction. The qualitative behaviour of the curves shown here can be understood as follows: For small non-local interaction the Mott-like ( × ) 3 3 R30° phase is dominant and unmixed with energetically high-lying configurations. However, upon increasing the nonlocal interactions some of these high-lying configurations (in particular the 3 × 3 and ( × ) 2 3 2 3 R30° states) become lower in energy with respect to the ground state and for nonlocal interactions exceeding sim 0.34 even replace the (  ×  )  3 3 R30° configuration as the main contribution for the ground state (note that mixing of the different configurations is driven by the gain of kinetic energy, i.e. electron hopping).

Multiplet cluster calculations.
In order to simulate the Sn 4d core-level spectra (shown in Fig. 1) we employ full multiplet cluster calculations using the code introduced in ref. 63. As common practice for such cluster simulated spectra we estimated the strength of the core hole spin-orbit coupling (SOC), and the multipole part of the core-valence interaction from atomic Hartree-Fock calculations 64 . For valence and core SOC we use ζ 5p = 0.40 eV, and ζ 4d = 0.41 eV; for the multipole moments of the core valence interaction we use the Slater integrals F 1 = 0.46 eV, F 2 = 1.47 eV, and F 3 = 0.42 eV 65 . The monopole part of the core-hole valence interaction is (in combination with the onsite U of Sn) responsible for the relative shift of the three spectra and fixed by the overall width of the spectrum. The spectral functions are calculated with exact diagonalization in the cluster limit and broadened by convoluting with a Gaussian of width 0.37 eV.
Theoretical ARPES spectra. The theoretical ARPES spectrum (top-right and middle-left panels of Fig. 2) is computed as a weighted average of the spectra ( ) for the 3 symme- The relative weights λ α are determined from the cluster diagonalisation and the cPES spectra (see above). α N el denote the number of electrons per unit cell for each symmetry (respectively 1, 3, 2). In the top-right panel, this spectrum is broadened with a Gaussian distribution of mean deviation σ = 0.3 to account for ARPES uncertainties. The individual spectra are shown in the remaining three panels. The self-energy of the (  ×  )  3 3 R30° symmetry is obtained by MaxEnt analytical continuation of the imaginary-frequency impurity self-energy ω Σ ( ) i imp computed self-consistently through an EDMFT scheme 43,60 . We find this self-energy to be reminiscent of an atomic self-energy with a renormalized interaction given by the self-consistently computed effective impurity interaction ω ( = ) 0  as obtained from GW + DMFT. For the 3 × 3 symmetry, we take the atomic self-energy for the half-filled band, while for the empty and full bands we take the Hartree estimates for the self-energy: For the × 2 3 2 3 symmetry, we also take Hartree estimates: