Direct Observation of Orbital Hybridisation in a Cuprate Superconductor
Christian E. Matt D. Sutter A. M. Cook Y. Sassa M. Mansson O. Tjernberg L. Das M. Horio D. Destraz C. G. Fatuzzo K. Hauser M. Shi M. Kobayashi V. Strocov P. Dudin M. Hoesch S. Pyon T. Takayama H. Takagi O. J. Lipscombe S. M. Hayden T. Kurosawa N. Momono M. Oda T. Neupert Johan Chang
cond-mat.supr-con cond-mat.str-el




  

    
   
   
    

   
   
 Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
 Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland
  
  
       
     Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
   
  Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
  
   
  Department of Physics and Astronomy, Uppsala University, SE-75121 Uppsala, Sweden
   
  KTH Royal Institute of Technology, Materials Physics, SE-164 40 Kista, Stockholm, Sweden
   
  KTH Royal Institute of Technology, Materials Physics, SE-164 40 Kista, Stockholm, Sweden
  
   
     Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
     
      
     Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
     
      
     Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
     
      

   
     Institute of Physics, École Polytechnique Fedérale de Lausanne (EPFL), Lausanne CH-1015, Switzerland

  
   Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
 

 
 Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland
 
 
  
 Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland

  
 Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland

 
 Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland


 Diamond Light Source, Harwell Campus, Didcot OX11 0DE, UK.
 
 
 Diamond Light Source, Harwell Campus, Didcot OX11 0DE, UK.

      
Department of Advanced Materials, University of Tokyo, Kashiwa 277-8561, Japan


Department of Advanced Materials, University of Tokyo, Kashiwa 277-8561, Japan


Department of Advanced Materials, University of Tokyo, Kashiwa 277-8561, Japan
      
    
         H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom
         
     
       H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom
     
     
Department of Physics, Hokkaido University - Sapporo 060-0810, 
Japan
 

Department of Physics, Hokkaido University - Sapporo 060-0810, 
Japan
Department of Applied Sciences, Muroran Institute of Technology, 
Muroran 050-8585, Japan


Department of Physics, Hokkaido University - Sapporo 060-0810, 
Japan

  
  Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
  
  
    Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland


Direct Observation of Orbital Hybridisation in a Cuprate Superconductor 
    J. Chang
    December 30, 2023
========================================================================





 



The minimal ingredients to explain the essential physics of  layered copper-oxide (cuprates) materials remains heavily debated.

Effective low-energy single-band models of the copper-oxygen orbitals are widely used

because there exists no strong experimental evidence supporting multi-band structures.
 
Here we report angle-resolved photoelectron spectroscopy experiments on 
La-based cuprates that provide direct observation of a two-band structure.
This electronic structure, qualitatively consistent with density functional theory, 
is parametrised by a two-orbital ( and ) tight-binding model. 
We quantify the orbital hybridisation  which  provides 
an explanation for the Fermi surface topology and the proximity of the van-Hove singularity to the Fermi level.
Our analysis leads to a unification of electronic hopping parameters for single-layer cuprates and we conclude that hybridisation, restraining d-wave pairing, is an important optimisation element for superconductivity. 





 








 


 
Identifying the factors that limit the transition temperature  of high-temperature cuprate superconductivity is a crucial step towards revealing the design principles underlying the pairing mechanism <cit.>. 
It may also provide an explanation for the dramatic variation of  across the 
known single-layer compounds <cit.>. Although superconductivity is certainly 
promoted within the copper-oxide layers,  
the apical oxygen position may play an important role in defining the transition temperature <cit.>. The CuO_6 octahedron lifts the degeneracy of the nine copper 3d-electrons and generates fully occupied t_2g and 3/4-filled e_g states <cit.>. With increasing apical oxygen distance  to the CuO_2 plane, the e_g states split to create a 1/2-filled  band. 
The distance  thus defines whether single or two-band models are most appropriate to describe the low energy band structure.
It has also been predicted that  influences  in at least two different ways.  First, the distance  controls the charge transfer gap between the oxygen and copper site which, in turn, suppresses superconductivity <cit.>.  Second, Fermi level  -hybridisation, depending on , reduces the pairing strength <cit.>. Experimental evidence, however, points in opposite directions. Generally, single layer materials with larger  have indeed a larger  <cit.>. However, STM studies of Bi-based cuprates suggest an anti-correlation between  and  <cit.>. 



In the quest to disentangle these causal relation between  and , it is imperative to experimentally 
reveal the orbital character of the cuprate band structure. 
The comparably short apical oxygen distance  makes   an ideal candidate for such a study.
Experimentally, however, it is challenging to determine the orbital character of the states near the Fermi energy (). In fact, the  band 
has never been identified directly by  angle-resolved photoelectron spectroscopy (ARPES) experiments. 

A large majority of ARPES studies have focused on the pseudogap, superconducting gap and quasiparticle self-energy properties in near vicinity to the Fermi level <cit.>. 

An exception to this trend are studies of the so-called waterfall structure <cit.> that lead to the observation of band structures below the  band <cit.>. However, the origin and hence orbital character of these bands was never addressed. Resonant inelastic x-ray scattering has been used to probe excitations between orbital d-levels. In this fashion, insight about the position of , , ,  states with respect to  has been obtained <cit.>. Although difficult to disentangle, it has been argued that for LSCO the  level is found above ,  ,  <cit.>. 


To date, a comprehensive study of the  momentum dependence is missing and therefore the coupling between the  and  bands has not been revealed. 

X-ray absorption spectroscopy (XAS) experiments, sensitive to the unoccupied states, concluded only marginal hybridisation of  and  states in (LSCO) <cit.>.
Therefore, the role of -hybridisation remains ambiguous <cit.>.      



Here we provide direct ultra-violet and soft-xray ARPES measurements 

of the  band in 
La-based single layer compounds. 
The  band is located about 1 eV below the Fermi level at the Brillouin zone (BZ) corners.  From these corners, the  band is dispersing downwards along the nodal and anti-nodal directions, consistent with density functional theory (DFT) calculations.

The experimental and DFT band structure, including only  and  orbitals, is parametrised using a two-orbital tight-binding model<cit.>.
The presence of the  band close to the Fermi level allows to describe the Fermi surface topology for all single layer compounds (including  and ) with similar hopping parameters for the  orbital. This unification of electronic parameters implies that the main difference between  single layer cuprates originates from  the hybridisation between  and  orbitals.
The significantly increased hybridisation in La-based cuprates pushes the van-Hove singularity close to the Fermi level. 
This explains why the Fermi surface differs from other single layer compounds. 
We directly quantify  the orbital hybridisation that plays a sabotaging role for superconductivity. 
 

  


 
Results


Material choices: Different dopings of  spanning from x=0.12 to 0.23 in addition 
to an overdoped compound of  with x=0.21 have been studied. 
These compounds represent different crystal structures:
low-temperature orthorhombic (LTO), low-temperature tetragonal (LTT)
and the high-temperature tetragonal (HTT).  Our results are very 
similar across all crystal structures and dopings (see Supplementary Fig. 1).  To keep the comparison 
to band structure calculations simple, this paper focuses on results obtained in 
the tetragonal phase of overdoped  with x=0.23.
 






Electronic band structure: A raw ARPES energy distribution map (EDM), along the nodal direction, is displayed in Fig. <ref>a.

Near , the widely studied nodal quasiparticle dispersion with predominately   character is observed <cit.>.  This band reveals the previously reported electron-like Fermi surface of , x=0.23<cit.> (Fig. <ref>b), the universal nodal Fermi velocity v_F≈1.5 eVÅ<cit.> and a band dispersion kink around 70 meV<cit.>. 
The main observation reported here is the second band dispersion at ∼1 eV below the Fermi level E_F (see Fig. <ref> and <ref>) and a hybridisation gap splitting the two (see Fig. <ref>).
 This second band – visible in both raw momentum distribution curves (MDC) 
and constant energy maps (CEM) – disperses downwards away from the BZ-corners. 
Since a pronounced k_z-dependence is observed  for this band structure (see Figs. <ref> and <ref>) a trivial surface state can be excluded . 
Subtracting a background intensity profile (see Supplementary Fig. 2) is a standard method that enhances visualisation of this second band structure. 
In fact, using soft x-rays (160 eV - 600 eV),  at least two additional bands (β and γ) are found below the  dominated 
band crossing the Fermi level. 
Here, focus is set entirely on the β band dispersion  closest to the  dominated band. 
This band is clearly observed at the BZ corners (see Figs. 1–3). The complete in-plane (k_x,k_y) and out-of-plane (k_z) band dispersion is presented in Fig. <ref>. 




 







 
Orbital band characters: To gain insight into the orbital character of these bands, a comparison with a DFT band structure 
calculation (see methods section) of La_2CuO_4  is shown in Fig. <ref>.
The e_g states ( and ) are generally found above the t_2g bands (, , and ).

The overall agreement between the experiment and 
the DFT calculation (see Supplementary Fig. 3) thus suggests that the two bands nearest to the Fermi level are composed 
predominately of  and  orbitals. This conclusion can also be reached by pure 
experimental arguments.


Photoemission matrix element selection rules contain information about the orbital band character. 
They can be probed in a particular experimental setup where a mirror-plane is defined by the incident light and the electron analyser slit <cit.>. 
With respect to this plane the electromagnetic light field 
has odd (even) parity for  () polarisation (see Supplementary Fig. 4).

Orienting the mirror plane along the nodal direction (cut 1 in Fig. <ref>), the  and  () orbitals have even (odd) parity.
For a final-state with even parity, selection rules <cit.> dictate that the  and -derived bands should appear (vanish) in the  () polarisation channel and vice versa for . Due to their orientation in real-space, the   and   orbitals are not expected to show a strict switching behavior along the nodal direction <cit.>.
As shown in Fig. <ref>f-g, two bands (α and γ) appear with -polarised light while for -polarised light bands β and γ' are observed.
Band α which crosses  is assigned to  while band γ has to originate from / orbitals as  and -derived states are fully suppressed for -polarised light. In the EDM, recorded with -polarised light, band (β) at ∼ 1 eV binding energy and again a band (γ') at ∼ 1.6 eV is observed.  
From the orbital shape, a smaller k_z dispersion is expected for  and -derived bands than for those from  orbitals.
As the β band exhibits a significant k_z dispersion (see Fig. <ref>), much larger than observed for 
 the  band, we conclude that it is of  character. The γ' band which is  very close to the γ band is  therefore of  character.
  Interestingly, this -derived band has stronger in-plane than out-of-plane dispersion, 
 suggesting that there is a significant 
  hopping to in-plane p_x and p_y oxygen orbitals.
 
 
 
 
Therefore the assumption that  the  states are probed uniquely through the apical oxygen p_z orbital <cit.> has to be taken with caution. 








Discussion

Most minimal models aiming to describe the cuprate physics start with an approximately 
half-filled  single  band on 
a two-dimensional square lattice.
Experimentally, different band structures have been observed  across single-layer cuprate compounds.
The Fermi surface topology of 
  is, for example, less rounded compared to  (Bi2201),  (Tl2201), and  (Hg1201).  
 Within a single-band tight-binding model the rounded Fermi surface shape of the single layer compounds Hg1201 and Tl2201 is  described by setting r=(|t_α'|+|t_α”|)/t_α∼0.4<cit.>, where t_α, t_α', and t”_α are nearest, next nearest and next-next nearest neighbour hopping parameters (see Table <ref> and Supplementary Fig. 4). 
For  with more flat Fermi surface sections, 
significantly lower values of r have been reported. For example, for overdoped , 

 r ∼0.2 was found <cit.>. 
The single-band premise thus leads to varying hopping parameters across the cuprate families,
 stimulating the empirical observation that T_c^max roughly scales with t_α' <cit.>.
This, however, is in direct contrast to t-J models that predict the opposite correlation <cit.>.
Thus the single-band structure applied broadly to all single-layer cuprates lead to conclusions that challenge
conventional theoretical approaches. 
 




  
The observation of the  band calls for a re-evaluation of the electronic structure
in La-based cuprates using a two-orbital tight-binding model (see methods section).
Crucially, there is a hybridisation term Ψ(k)=2 t_αβ[cos( k_x )-cos(k_y)] between the  and  orbitals, where t_αβ is a hopping parameter that characterises the strength of orbital hybridisation. 
In principle, one may attempt to describe the two observed bands 
independently by taking t_αβ=0. However, the problem then
returns to the single band description with the above mentioned contradictions. 
Furthermore, t_αβ=0 implies a band
crossing in the antinodal direction that is not observed experimentally (see Fig. <ref>). In fact, from the 
avoided band crossing one can directly estimate t_αβ≈200 meV. 


As dictated by the different eigenvalues of the orbitals under mirror symmetry, the hybridisation term Ψ(k) vanishes on the nodal lines k_x=± k_y (see inset of Fig. 3). Hence the pure  and  orbital band character is expected along these nodal lines. 

The hybridisation Ψ(k) is largest in the anti-nodal region, pushing the van-Hove singularity of the upper band close to the Fermi energy and in case of overdoped  across the Fermi level. 

 


In addition to the hybridisation parameter t_αβ and the chemical potential μ,
six free parameters 
enter the tight-binding model that yields the entire band structure (white lines in Figs. <ref> and <ref>). Nearest and next-nearest in-plane hopping
parameters between  (t_α, t_α') and  (t_β,t_β') orbitals are introduced to capture the 
Fermi surface topology and in-plane  band dispersion (see Supplementary Fig. 4). The k_z dispersion is described by nearest and next-nearest out-of-plane hoppings (t_β z, t'_β z) of the  orbital. 
The four  hopping parameters and the chemical potential μ are determined from the experimental band structure
 along the nodal direction where Ψ(k)=0. Furthermore, the  α and β band dispersion in the anti-nodal region and the Fermi surface topology provide the parameters t_α, t_α' and t_αβ. 
 Our analysis 
 reveals a finite band coupling  -t_αβ=0.21 eV 
 resulting in a strong anti-nodal orbital hybridisation (see Fig. <ref> and Table <ref>).
 
Compared to the single-band parametrisation <cit.> a significantly larger value r ∼ -0.32  is found 
and hence a unification of t_α'/t_α ratios for all single-layer compounds is achieved.






Finally, we discuss the implication of orbital hybridisation for superconductivity and pseudogap physics.
First, we notice that a pronounced pseudogap is found in the anti-nodal region 
of  with x=0.21 – consistent with transport experiments <cit.> (see Supplementary Fig. 5).  The fact that t_αβ of La_1.59Eu_0.2Sr_0.21CuO_4 is similar to  t_αβ of LSCO suggests that the pseudogap is not suppressed 

by the  -hybridisation. 
To this end, 

a comparison to the 1/4-filled e_g system 
Eu_2-xSr_xNiO_4 with x=1.1 is interesting <cit.>.
This material has the same two-orbital band structure with protection against hybridisation along 
the nodal lines. Both the  and  bands are crossing the Fermi level, producing two Fermi surface sheets <cit.>. 
Despite an even stronger  -admixture of the  derived band a  d-wave-like pseudogap has been reported <cit.>. The pseudogap physics 
thus seems to be unaffected by the orbital hybridisation. 

It has been argued that orbital hybridisation – of the kind reported here – is unfavourable for superconducting pairing<cit.>.




It thus provides an explanation for the  varying T_c^max across single layer 
cuprate materials. Although other mechanisms, controlled by the apical oxygen distance, (e.g. variation of the copper-oxygen charge transfer gap <cit.>) are not excluded our results demonstrate that orbital hybridisation exists and is an important control parameter for superconductivity.  





Acknowledgements:
 D.S., D.D., L.D., and J.C. acknowledge support by the Swiss National Science Foundation. Further, Y.S. and M.M. are supported by the Swedish Research Council (VR) through a project (BIFROST, dnr.2016-06955).
This work was performed at the SIS<cit.>, ADRESS <cit.> and I05 beamlines at the Swiss Light Source and at the Diamond Light Source. A.M.C. wishes to thank the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293, for hosting during some stages of this work. 


We acknowledge Diamond Light Source for access to beamline I05 (proposal SI10550-1) that contributed to the results presented here  and thank all the beamline staff for technical support. 



Authors contributions:
SP, TT, HT, TK, NM, MO, OJL,  and SMH grew and prepared single crystals. 
CEM, DS, LD, MH, DD, CGF, KH, JC, NP, MS, OT, MK, VS, TS, PD, MH, MM, and YS prepared and carried out the ARPES experiment.
CEM, KH, JC performed the data analysis. CEM carried out the DFT calculations and 
AMC, CEM, TN developed the tight-binding model.
All authors contributed to the manuscript. 


Methods 

Sample characterisation: High-quality single crystals of , x=0.12, 0.23, and , x=0.21, were grown by the  floating-zone technique. 
The samples were characterised by SQUID magnetisation <cit.> to determine superconducting transition temperatures (T_c=27 K, 24 K and 14 K). 
For the crystal structure, the experimental lattice parameters are a=b=3.78 Åand c=2c'=13.2 Å<cit.>. 



ARPES experiments: Ultraviolet and soft x-ray ARPES experiments were carried out at the SIS  and ADRESS beam-lines at the Swiss Light Source and at the I05 beamline at Diamond Light Source. Samples were pre-aligned ex situ using a x-ray LAUE instrument and cleaved  in situ – at base temperature (10 - 20 K) and ultra high vacuum (≤5·10^-11 mbar) – employing a top-post technique or cleaving device<cit.>. 
Ultraviolet (soft x-ray <cit.>) ARPES spectra were recorded using a SCIENTA R4000 (SPECS PHOIBOS-150) electron analyser with horizontal (vertical) slit setting. All data was recorded at the cleaving temperature 10-20 K. To visualize the -dominated band, we subtracted in Figs. <ref>f,g and Figs. <ref>-<ref> the background 
that was obtained by taking the minimum intensity of the MDC at each binding energy. 


Tight-binding model:

A two-orbital tight-binding model Hamiltonian 

with symmetry-allowed hopping terms is employed to isolate and characterise the extent of orbital hybridisation of the observed band structure <cit.>. For compactness of the momentum-space Hamiltonian matrix representation, 

we introduce the vectors


    Q^κ_   = ( a, κ   b,0)^𝖳, 
    R^κ_1,κ_2   =  (κ_1 a , κ_1κ_2  b ,c )^𝖳 /2, 
    T^κ_1,κ_2_1    = (3κ_1 a , κ_1κ_2  b ,c )^𝖳/2, 
    T^κ_1,κ_2_2    = (κ_1 a ,3 κ_1κ_2  b ,c )^𝖳/2,




where 
κ, κ_1, and κ_2 take values ±1 as defined by sums in the Hamiltonian and 𝖳 denotes vector transposition.  


Neglecting the electron spin (spin orbit coupling is not considered) the momentum-space tight-binding Hamiltonian, H(k), at a particular momentum k = (k_x, k_y, k_z ) 

is then given by


    H(k) = [ [ M^x^2-y^2(k)         Ψ(k);         Ψ(k)     M^z^2(k) ]]


in the basis 

(c_k, x^2 - y^2, c_k, z^2)^⊤, 
where the operator c_k,α annihilates an electron with momentum k 

in an e_g-orbital d_α, with α∈{ x^2-y^2, z^2 }. 
The diagonal matrix entries

are given by

    M^x^2-y^2(k) =    2t_α[ cos( k_x a) + cos( k_y b ) ] + μ
       + ∑_κ=±1 2 t_α' cos(Q^κ_·k) 
       + 2t_α”[ cos( 2k_x a) + cos( 2k_y b ) ],


and

    M^z^2(k) =       2 t_β[ cos ( k_x a) + cos ( k_y b) ] 
    -μ
       +  ∑_κ = ± 12 t_β'cos( Q^κ_·k)  
       +∑_κ_1,2 = ± 1[ 2 t_β zcos( R_^κ_1,κ_2·k)  
       + ∑_i=1,2 2 t'_β zcos(T^κ_1,κ_2_i ·k) ],


which describe the intra-orbital hopping for  and  d_z^2 orbitals, respectively.
The inter-orbital nearest-neighbour hopping term is given by 


    Ψ(k)    = 2 t_αβ[ cos( k_x a ) - cos( k_y b ) ].

 
In the above, μ determines the chemical potential.

The hopping parameters ,  and  characterise nearest neighbor (NN), next-nearest neighbour (NNN) and next-next-nearest neighbour (NNNN) intra-orbital in-plane hopping between  orbitals.  and  characterise NN and NNN intra-orbital in-plane hopping between  orbitals, while  and  characterise NN and NNN intra-orbital out-of-plane hopping between  orbitals, respectively (see Supplementary Fig. 3). Finally, the hopping parameter  characterises NN inter-orbital in-plane hopping. 
Note that in our model,  intraorbital hopping terms described by the vectors (Eqs. <ref>) are neglected as these  are expected to be weak compared to those of the  orbital.

This is due to the fact that the inter-plane hopping is mostly mediated by hopping between apical oxygen p_z orbitals, which in turn only hybridize with the  orbitals, not with the  orbitals. Such an argument highlights that the tight-binding model is not written in atomic orbital degrees of freedom, but in Wannier orbitals, which are formed from the Cu d orbitals and the ligand oxygen p orbitals. As follows from symmetry considerations and is discussed in Ref. HirofumiPRB12, the Cu  orbital together with the apical oxygen p_z orbital forms a Wannier orbital with  symmetry, while the Cu  orbital together with the four neighboring p_σ orbitals of the in-plane oxygen forms a Wannier orbital with  symmetry. One should thus think of this tight-binding model as written in terms of these Wannier orbitals, thus implicitly containing superexchange hopping via the ligand oxygen p orbitals.


Additionally we stress that all hopping parameters 
effectively include the oxygen orbitals.

Diagonalising Hamiltonian (<ref>), we find two bands 


    ε_±(k) =    1 2[ M^x^2-y^2(k) +  M^z^2(k) ] 
       ±1  2√([M^x^2-y^2(k ) - M^z^2(k )  ]^2 + 4 Ψ^2 (k ) )


and make the following observations: along the  k_x=± k_y lines, Ψ(k) vanishes and hence no orbital mixing appears in the nodal directions. 
The reason for this absence of mixing lies in the different mirror eigenvalues of the two orbitals involved. Hence it is not an artifact of the finite range of hopping processes included in our model. The parameters of the tight-binding model are determined by fitting the experimental bandstructure and are provided in Table <ref>. 



 










 DFT calculations: 
Density functional theory (DFT) calculations were performed for La_2CuO_4 in the tetragonal space group I4/mmm, No. 139, found in the overdoped regime of  using the WIEN2K package <cit.>. 
Atomic positions are those inferred 
from neutron diffraction measurements<cit.> for x=0.225. 
In the calculation, the Kohn-Sham equation is solved self-consistently by using a full-potential linear augmented plane wave (LAPW) method. The self consistent field calculation converged properly for a uniform k-space grid in the irreducible BZ. The exchange-correlation term is treated within the generalized gradient approximation (GGA) in the parametrization of Perdew, Burke and Enzerhof (PBE)<cit.>. The plane wave cutoff condition was set to RK_max=7 where R is the radius of the smallest LAPW sphere (i.e., 1.63 times the Bohr radius) and K_max denotes the plane wave cutoff. 

    
 



Data availability.

 All experimental data are available upon request to the corresponding authors.





Competing financial interest.

The authors declare no competing financial interests.



Additional information.
Correspondence to: J. Chang (johan.chang@physik.uzh.ch) and C. Matt (cmatt@g.harvard.edu).





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