# Theoretical prediction of a strongly correlated Dirac metal

I. I. Mazin  Code 6393, Naval Research Laboratory, Washington, DC 20375, USA
Harald O. Jeschke  Institut für Theoretische Physik, Goethe-Universität
Frankfurt, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany  Frank
Lechermann  I. Institut für Theoretische Physik, Universität Hamburg, D-20355
Hamburg, Germany  Hunpyo Lee  Institut für Theoretische Physik, Goethe-
Universität Frankfurt, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main,
Germany  Mario Fink  Institut für Theoretische Physik I, Universität Würzburg,
am Hubland, 97074 Würzburg, Germany  Ronny Thomale  Institut für Theoretische
Physik I, Universität Würzburg, am Hubland, 97074 Würzburg, Germany  Roser
Valentí  Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-
von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany

###### Abstract

Recently, the most intensely studied objects in the electronic theory of
solids have been strongly correlated systems and graphene. However, the fact
that the Dirac bands in graphene are made up of $sp^{2}$-electrons, which are
subject to neither strong Hubbard repulsion $U$ nor strong Hund’s rule
coupling $J$ creates certain limitations in terms of novel, interaction-
induced physics that could be derived from Dirac points. Here we propose
GaCu3(OH)6Cl2 (Ga-substituted herbertsmithite) as a correlated Dirac-Kagome
metal combining Dirac electrons, strong interactions and frustrated magnetism.
Using density functional theory (DFT), we calculate its crystallographic and
electronic properties, and observe that it has symmetry-protected Dirac points
at the Fermi level. Its many-body physics is excitingly rich, with possible
charge, magnetic and superconducting instabilities. Through a combination of
various many-body methods we study possible symmetry-lowering phase
transitions such as Mott-Hubbard, charge or magnetic ordering, and
unconventional superconductivity, which in this compound assumes an $f$-wave
symmetry.

###### pacs:

74.62.Dh,75.10.Kt,81.05.Zx

Introduction. It is well known theoretically that graphene is not the only
system where crystallography, combined with a particular one-electron
Hamiltonian, creates symmetry-protected Dirac points at a certain filling. As
such, the hope persists to combine features of Dirac fermions and strong
correlations in an experimentally accessible condensed matter system. For
instance, the single-orbital kagome tight-binding model is known to feature,
besides a flat band that has been subject of considerable interest, symmetry-
protected Dirac points at a filling of $n=4/3$ electrons per site (or,
equivalently, $1/3,$ for reversed sign of the hopping). However, so far this
fact has been considered more a numerical curiosity of a simplified model than
an accessible feature in real materials, and has received little attention.
Currently, the predominantly studied kagome material has been herbertsmithite
Shores2005 ; Lee2008 ; Mendels2010 , ZnCu3(OH)6Cl2, presumed to host a spin
liquid with fractionalized spin excitations, consistent with recent neutron
scattering measurements herbert-nature . It might be the first manifestation
of a truly two-dimensional resonating valence bond (RVB) spin liquid
anticipated in theory 27 years ago ANDERSON06031987 ; PhysRevB.35.8865 ;
PhysRevB.44.2664 ; klein1991resonating , which is supported by large-scale
numerical investigations Yan03062011 ; PhysRevLett.109.067201 ;
PhysRevB.89.020408 .

Herbertsmithite Jeschke2013  is the starting point of our proposal to unify
strong correlations, metallicity and Dirac fermions. As opposed to the flat
band which only exists in the nearest neighbour single-orbital tight binding
model, the Dirac points in the kagome compounds are symmetry-protected and
survive in any electronic structure model that respects the hexagonal
symmetry. By replacing Zn2+ by Ga3+ in herbertsmithite, GaCu3(OH)6Cl2 (Fig.
1), the Dirac points are placed at the Fermi level. These states are formed by
the strongly correlated ($U/t\sim 50$) Cu$-d_{x^{2}-y^{2}}$ orbitals, whose
nearest neighbour hopping Hamiltonian on the kagome lattice appears to be
equivalent to that of $s$-orbitals. As we will further elaborate on below, we
find a strong, compared to graphene, screening, hinting at rather local
interactions, while the nature of magnetic interactions is typical for the
kagome lattice, suggesting no long range antiferromagnetism and a flat spin
fluctuation profile.

Zn and Ga are neighbours in the periodic table and their radii are similar.
Even if they do not form a continuous solid solution, our calculations suggest
that the fully substituted compound, GaCu3(OH)6Cl2, should be dynamically
stable, and it should be possible to dope it with holes by partially replacing
Ga with Zn. If the solid solution Zn1-xGaxCu3(OH)6Cl2 were to exist, it would
span the whole range from a fully frustrated RVB spin liquid ($x=0$) at half
filling $n=1$ to a strongly correlated Dirac metal ($x=1$) at $n=4/3$. The
ground state in the former limit is known to be a uniform Mott insulator. Upon
doping, a number of instabilities can manifest themselves, and in fact one can
anticipate a complex and highly interesting phase diagram. Specifically, the
DFT noninteracting one-electron picture may be prone to (1) Mott-Hubbard
metal-insulator transition, (2) charge ordering, (3) ferromagnetism or (4) a
superconducting instability. Below we will first present our DFT calculations,
and then discuss these possible instabilities by means of various
complementary many-body model approaches.

First principles calculations. Herbertsmithite is part of the Zn-paratacamite
family of compounds Mendels2010  that shows great flexibility in its
composition both naturally (e.g., Ni-herbertsmithite Clissold2007 ) and
synthetically (e.g. Mg-herbertsmithite Colman2011 ). While Ga more commonly
assumes a fourfold coordination, it also occurs in the sixfold one Donnay_1971
. In the framework of DFT we fully relaxed the herbertsmithite structure after
replacing Zn by Ga (see the Supplementary Information).

The resulting electronic structure (Fig. 2 b) shows low-energy bands
qualitatively similar to a one-orbital model on the kagome lattice (Fig. 2 a)
and has Dirac points exactly at the Fermi level. In order to understand this
fact, we first observe that Cu in this structure sits inside an oxygen square,
so that the $d_{x^{2}-y^{2}}$ orbitals, with strong $pd\sigma$ hopping to the
oxygens, form a high-lying antibonding band (Fig. 3). If one neglects the
other $d$ orbitals, and integrates out oxygens, one gets a tight-binding
model, which is mathematically equivalent to a single $s$-orbital model with
the nearest neighbour hopping $t=t_{pd\sigma}^{2}/(E_{F}-E_{p})>0.$ The
solution of this model is well known (Fig. 2 a). The bands crossing along the
$\Gamma-K$ line have different parity with respect to the $y/-y$
transformation ($y$ $\perp\Gamma K$) , and therefore the Dirac points at K are
symmetry-protected for an arbitrary hopping range (apart from the very small
spin-orbit coupling).

The kagome tight-binding model has Dirac points at $K$; however, this is not
the case for a finite $k_{z}$ dispersion and rhombohedral symmetry (Fig. 2 c).
In a planar system with hexagonal symmetry one can travel from $\Gamma$ to $M$
and then continue to another $\Gamma$. In a rhombohedral structure, if one
travels along the same direction one actually ends up at Z, and the $k_{z}=0$
plane alone has lower symmetry than if averaged over all $k_{z}$. The Dirac
points remain nevertheless protected, but form a 3D line meandering about the
vertical line passing through the hexagonal K point (see Fig. S1 in the
Supplementary Information), which, if averaged over $k_{z}$, projects onto
this point. When doped, the Fermi surface consists of six tubes that slightly
twist as we move along $k_{z}$ (Fig. 2 f, g).

Figure 1: Structure of predicted herbertsmithite modification GaCu3(OH)6Cl2. a
View of an individual kagome plane. b Full view along $c$ axis where three
shifted kagome layers are stacked. c Side view.  Figure 2: Electronic
structure of GaCu3(OH)6Cl2. a Band structure and b density of states of the
nearest-neighbor tight-binding model for the kagome lattice. c-g Electronic
structure of Ga-substituted herbertsmithite, GaCu3(OH)6Cl2. c Band structure
along high symmetry points of the hexagonal setting of the $R\bar{3}m$ space
group. d Density of states showing GaCu3(OH)6Cl2to be a zero gap
semiconductor. e Brillouin zones of the $R\bar{3}m$ space group in
rhombohedral (purple) and hexagonal setting (green), with the path chosen in c
(see also a view from the top in Fig. S1, Supplementary Information). f and g
Fermi surfaces at an energy $E=-60$ meV. The Fermi surface plots demonstrate
that the material is two-dimensional to a good approximation.  Figure 3:
Relevant tight-binding orbitals for GaCu3(OH)6Cl2. Overlap between Cu
$3d_{x^{2}-y^{2}}$ and O $p$ orbitals in a Cu triangle. $+$ and $-$ sign of
the wave function is coded with black and white, respectively.

Normal state instabilities.- Next we want to check the system against normal
state electronic instabilities. In a strongly correlated system this cannot be
adequately addressed within the DFT. Therefore, we have constructed a many-
body model Hamiltonian that captures the most important interactions:

$\begin{split}\mathcal{H}&=\mathcal{H}_{0}+\mathcal{H}_{\text{int}}\\\
\mathcal{H}_{0}&=\sum_{i,j}\sum_{\sigma}t_{ij}c_{i,\sigma}^{\dagger}c_{j,\sigma}+\mu\sum_{i,\sigma}c_{i,\sigma}^{\dagger}c_{i,\sigma}\\\
\mathcal{H}_{\text{int}}&=U\sum_{i}n_{i,\uparrow}n_{i,\downarrow}+\frac{V}{2}\sum_{<i,j>}\sum_{\sigma,\sigma^{\prime}}n_{i,\sigma}n_{j,\sigma^{\prime}}.\end{split}$
(1)

where the first term is the effective Cu $d_{x^{2}-y^{2}}$ tight-binding
Hamiltonian derived from the DFT band structure (see the Supplementary
Information) and the onsite $U$ and nearest-neighbor Coulomb repulsion $V$
between Cu $d_{x^{2}-y^{2}}$ atoms we estimated (see the Supplementary
Information) to be $U=5-7$ eV and $V=0.11$eV. It is worth noting that the
latter number is unexpectedly small. The reason for that is twofold: first,
even at the exact $n=4/3$ filling, where the metallic density of states (DOS)
is zero, the behaviour of the dielectric constant is nearly metallic, and the
Coulomb interaction is well screened; second, even though the electronic
structure is quasi-2D, it is a far cry from an isolated plane as in graphene —
each plane participates in screening interactions not only within the plane,
but in all other planes.

Mott-Hubbard instability.- ZnCu3(OH)6Cl2 is a Mott insulator since the Cu
$d_{x^{2}-y^{2}}$ state is half-filled and electron hopping incurs an energy
cost of $U.$ This is not the case in GaCu3(OH)6Cl2. Its filling of $n=4/3$
corresponds to one electron per site plus one extra electron per three sites.
Suppose $U=\infty.$ Then exactly 1/3 of all sites are double, and the rest are
single occupied. Thus, one electron per site will be localised and 1/3 of an
electron mobile, and this mobility will not be impeded at all. This simplistic
treatment suggests that the system will be metallic.

To verify that, we applied to the Hamiltonian Eq. 1 the dynamical cluster
approximation (DCA) within the rotationally invariant slave-boson, DCA(RISB),
formalism li89 ; lec07  in the saddle-point approximation (see the
Supplementary Information). As a minimal cluster we considered three sites.
Figure 4 shows our DCA(RISB) results in the paramagnetic regime. For $U=5$ eV,
we observe a band-narrowing but the essential low energy features, especially
the pseudogap reflecting the Dirac points, remain. Thus, the onsite repulsion
alone is insufficient to drive the system insulating at $n=4/3$. This is a key
result, and we have confirmed it by using the more accurate continuous–time
quantum Monte Carlo approach (see the Supplementary Information).

Charge ordering.- Having verified the absence of a Mott-Hubbard transition, we
turn to possible charge ordering, which in the limit $U\gg t$ is controlled by
the intersite Coulomb repulsion $V$. At the mean-field level, the competition
is between the energy gain of charge ordering, and the loss of kinetic energy.
The former can be estimated from the fact that in the charge ordered state no
neighbouring sites are doubly occupied, thus, the nearest neighbour repulsion
is completely avoided. On the other hand, if all electrons are randomly
distributed over all sites, each nearest neighbour bond has (1/3)2=1/9
probability to have double occupancy on both sides, losing $V/9$ per bond, or
$2V/9$ per site. The kinetic energy loss can be estimated as ($2/3-1/6)=1/2$
of the energy of non-interacting electrons on the kagome lattice
($E_{0}\approx-0.8t$ per site for $n=4/3$), comment_V . Comparing $2V/9$ and
$0.4t$ we conclude that the system may develop charge ordering at $V\gtrsim
2t.$ Our DFT calculations yield $t\sim 0.3$ eV, however, one may expect a
renormalisation due to correlation effects; a factor of two is reasonable for
Cu $d$ electrons and we get $V_{c}\sim t\sim 0.3$ eV. This value is much
larger than $V$ estimated for GaCu3(OH)6Cl2, and, therefore, it should remain
a uniform metal.

Again, we subjected these qualitative arguments to a numerical test solving
Eq. 1 within the DCA(RISB) for $U=5$ eV and different $V$ values. At a
critical $V_{c}=0.54$ eV, (Fig. 4 b,c) well above the estimated value of
$V\approx 0.11$ eV, the system charge orders and loses its metallicity,
consistent with the qualitative arguments above. Interestingly, close to and
inside the insulating phase the system shows substantially increased values of
the intersite terms in the quasi-particle weight (Fig. 4 d) as well as in the
cluster orbital density matrix (Fig. 4 e). Inspection of the slave-boson
multiplet amplitudes reveal an insulator composed of dominant coherent
superpositions of the four-particle triangular-cluster states with one double-
occupied site and two single-occupied sites with different spin orientation, a
reflection of proximity to charge ordering at $V_{c}=0.54$ eV.

Note that in our functional renormalisation group (fRG) calculations,
described below, the problem of the phase instabilities is approached from the
opposite limit, namely the itinerant/weak coupling limit, valid at $U\lesssim
t$, appropriate for addressing the superconductivity. It is worth mentioning
though, that in fRG the $V_{c}$ needed to stabilise the leading charge
ordering instability was also far greater than the realistic range of $V$.

Figure 4: DCA-like rotationally invariant slave boson results for
GaCu3(OH)6Cl2. a-b Quasi-particle (QP) spectral function for different
interactions on the three-site cluster. The curves are scaled with the
respective onsite QP weight$Z$. c-e cluster quantities with intersite $V$:
onsite QP weight c, intersite QP weight d, and intersite terms of the cluster
orbital density matrix e.

Ferromagnetism.- We consider next the possibility of a ferromagnetic
instability. It is well known that Hubbard models with very large $U/t$ and
away from half-filling show propensity to so-called Nagaoka ferromagnetism
Nagaoka . Existing calculationsHanisch97  suggest that GaCu3(OH)6Cl2 is far
from the Nagaoka regime. However, these works did not consider states with
partial spin-polarization, which are natural in a metallic system.

Let us first estimate the effective magnetic interactions at the mean-field
level by comparing the superexchange $J_{ex}\sim 4t^{2}/U\sim 70$ meV and the
ferromagnetic kinetic energy gain per bond, times the number of mobile
electrons per site. $E_{KE}=E_{0}/3\times 1/3\sim 30$ meV. $E_{KE}<J_{ex}$,
suggesting a spin-liquid without a long range order. However, this estimate
also does not account for partial polarization, and in fact shows that the two
competing interactions are of the same order of magnitude. Therefore we looked
for ferromagnetism with the DCA(RISB), and found no ferromagnetism at $n=4/3$
(Dirac metal) and $n\lesssim 4/3$ (hole-doping) for any reasonable parameters.
In contrast, with electron doping ($n>4/3$), as the superexchange is gradually
suppressed, ferromagnetism becomes increasingly more favourable.

The importance of electron itineracy is particularly clear from our weak
coupling fRG calculations (see the Supplementary Information). By
construction, fRG shows no instabilities for zero DOS, but as we dope away
from the Dirac point (either holes or electrons), we find, at sufficiently
large $U$, a ferromagnetic instability. It should be noted that fRG depends on
resummation of all parquet diagrams, an approximation not rigorously justified
when $U$ exceeds the band width. As we do not see clean divergent RG flow for
the ferromagnetic channel, we cannot distinguish ferromagnetic fluctuations
from a partially-polarised ferromagnet. From the combined view of different
methodolgies, however, we can ascertain that the system is very close to a
ferromagnetic instability; whether it is realized in GaCu3(OH)6Cl2, we cannot
be sure. But, importantly, strong ferromagnetic fluctuations are essential for
unconventional superconductivity, discussed in the next section.

Superconductivity.- Finally, we investigate the possibility of
superconductivity in GaCu3(OH)6Cl2. Obviously, when the Fermi level is exactly
at the Dirac point the effective DOS (averaged over the cutoff energy) is
small, of the order of $J_{ex}v_{F},$ where the cutoff is set at the exchange
energy; $v_{F}$ is the Dirac velocity lambda . However if the Ga substitution
is incomplete and the Fermi surfaces form narrow cylinders (Fig. 2 f,g) one
expects sizeable coupling with spin excitations. This can be qualitatively
estimated in the simple model of Fig. 2 a as follows: The pairing interaction
can be written, roughly, as $t^{2}S(q,\omega),$ where $S(q,\omega)$ is the
Fourier-transform of the spin-spin correlator $\left\langle
S_{i}S_{j}\right\rangle.$ Assuming that only the nearest spins are correlated
(antiferromagnetically), we get three branches, one corresponding to an
uncorrelated spin liquid, $S(q,\omega)=0,$ and two dispersive ones. As
discussed in Ref. PhysRevB.45.2899, , this corresponds to three magnon modes,
one dispersionless and two acoustic; the corresponding pairing interaction is
strong at $q\sim 0$ and weakens for larger $q$ values, and is repulsive for
singlet and attractive for triplet pairing. Note that possible ferromagnetic
fluctuations, not included in this simplistic picture, are $also$ peaked at
$q\sim 0$ and thus the arguments above apply to them, too.

Now we observe that in a hexagonal Brillouin zone the 6 K-points related by
the rotational symmetry (Supplementary Information, Fig. S1) can be separated
into two subsets, $K$ and $K^{\prime},$ such that all points of a given subset
are also connected by translational symmetry. Given that the superconducting
order parameter has to honor translational symmetry (but not necessarily
rotational one), we see that as we go around the Brillouin zone the phase
factors between the consecutive pockets (neglecting intrapocket variations)
vary as $\phi,$ $\phi^{\prime},$ $\phi,$ $\phi^{\prime},$ $\phi,$
$\phi^{\prime},$ corresponding to an $L=3$ ($f$-wave) harmonic. Indeed there
exists a state RevModPhys.63.239 , often labelled $B_{1u}$, with the order
parameter transforming as $\mathbf{\hat{z}}x(x^{2}-3y^{2}),$ so that
$\phi=-\phi^{\prime}.$ In this case the intrapocket ($e.g.$, small $q$)
interaction is attractive and pairing, while the interpocket one is repulsive
and pairbreaking. However, the interpocket interaction is suppressed (assuming
an acoustic spectrum) as ($k_{F}/G)^{2}$, with the distance $G=K-K^{\prime}$.
One can estimate the BCS $T_{c}$=$\omega\exp(-1/\lambda),$ where $\omega\sim
Jk_{F}a$ (Ref. PhysRevB.45.2899, , Eq. 2.23), and $\lambda\sim
t^{2}N(E_{F})/\omega\sim t/J.$ We see that (i) the coupling constant is
roughly independent of doping, (ii) the prefactor changes approximately as a
square root of doping and (iii) one can expect a sizeable $T_{c}$ for
$k_{F}\lesssim G/2$ (the maximum possible value before changing the topology).
For $\lambda\sim 1,$ which seems to be a conservative estimate, and $k_{F}\sim
G/4,$ we get $T_{c}\sim 0.07J\sim 30$ K for $J\sim 40$ meV. Note that in this
regime dependence on $\lambda$ is close to linear. Taking $\lambda\sim 2$
raises the estimate for $T_{c}$ to 60 K. Of course, all these estimates are
order of magnitude at best, therefore we turn again to a quantitative
treatment in terms of multi-sublattice fRG pht . Note that superconductivity
is an itinerant effect, and the inherent fRG assumption of weak coupling
should be acceptable.

Around $n=4/3$ and constraining ourselves to the superconducting channel,
which, from a Kohn-Luttinger perspective, would become the leading channel for
sufficiently weak interactions, we find a clear preference for nodeless
$f$-wave superconductivity, as anticipated from the qualitative argument
above. The calculated gaps, while changing sign between the pockets, remain
uniform inside each pocket (see Supplementary Information, Fig. S3). It is
worth noting that the disorder introduced by Zn doping, while weak, might
even, counterintuitively, support superconductivity in this
limitPhysRevB.87.174511 . Finally, a subtlety could arise for significant $V$:
In order to avoid onsite and nearest neighbour repulsion, this could drive a
transition from a nodeless $B_{1u}$ $f$-wave to nodal $B_{2u}$ $f$-wave Ronny
, but for realistic values of $V$ this is highly unlikely.

Conclusions.- This work opens many interesting avenues toward novel physics in
electron-doped herbertsmithite compounds. Bridging between an RVB scenario of
frustrated magnetism for $x=0$ and a correlated Dirac metal at $x=1$,
$\text{Zn}_{1-x}\text{Ga}_{x}$Cu3(OH)6Cl2 promises to exhibit a plethora of
unconventional electronic phases as a function of doping, temperature, and
disorder. Even if a solid solution $0<x<1$ (which would be an ultimate bonanza
of novel physics) would not be realized experimentally, pure GaCu3(OH)6Cl2
($x=1)$ and its slightly hole doped version ($x\lesssim 1)$ should be highly
interesting systems, and can still be characterized as doped RVB. According to
our analysis, in this regime (close to the Dirac points) the two leading
candidates for the low-temperature ground state are weak ferromagnetism and
$f$-wave superconductivity. Further experimental and theory work shall decide
who is the winner in their competition.

Acknowledgements

We acknowledge useful discussions with C. Krellner, C. Platt, G. Khalliulin,
A. Chubukov and C. Piefke. I.I.M. is supported by ONR through the NRL basic
research program. H.O.J., F.L. and R.V. are supported by DFG-FOR1346 and DFG-
SFB/TR49 (H.O.J., R.V.). R.T. is supported by the European Research Council
through the grant TOPOLECTRICS, ERC-StG-336012.

Author contributions Ab initio density functional calculations, conceptual
development and analysis: I.I.M., H.O.J., R.V. DCA(RISB) calculations: F.L.
DCA(CT-QMC) calculations H.L. fRG calculations: M.F, R.T. All authors
participated in writing the manuscript.

## References

  * (1) Shores, M. P., Nytko, E. A., Bartlett, B. M. & Nocera, D.  A structurally perfect $S=1/2$ kagome antiferromagnet.  _J. Am. Chem. Soc._ 127, 13462 (2005).
  * (2) Lee, P. A.  An end to the drought of quantum spin liquids.  _Science_ 321, 1306–1307 (2008).
  * (3) Mendels, P. & Bert, F.  Quantum kagome antiferromagnet ZnCu3(OH)6Cl2.  _J. Phys. Soc. Jpn._ 79, 011001 (2010).
  * (4) Han, T.-H. _et al._ Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet.  _Nature_ 492, 406 (2012).
  * (5) Anderson, P. W.  The resonating valence bond state in La2CuO4 and superconductivity.  _Science_ 235, 1196–1198 (1987).
  * (6) Kivelson, S. A., Rokhsar, D. S. & Sethna, J. P.  Topology of the resonating valence-bond state: Solitons and high $T_{c}$ superconductivity.  _Phys. Rev. B_ 35, 8865–8868 (1987).
  * (7) Wen, X. G.  Mean-field theory of spin-liquid states with finite energy gap and topological orders.  _Phys. Rev. B_ 44, 2664–2672 (1991).
  * (8) Klein, D., Schmalz, T. G., García Bach, M. A., Valentí, R. & Živkovic, T. P.  Resonating-valence-bond theory for the square-planar lattice.  _Phys. Rev. B_ 43.
  * (9) Yan, S., Huse, D. A. & White, S. R.  Spin-liquid ground state of the $S=1/2$ kagome Heisenberg antiferromagnet.  _Science_ 332, 1173–1176 (2011).
  * (10) Depenbrock, S., McCulloch, I. P. & Schollwöck, U.  Nature of the spin-liquid ground state of the $S=1/2$ Heisenberg model on the kagome lattice.  _Phys. Rev. Lett._ 109, 067201 (2012).
  * (11) Suttner, R., Platt, C., Reuther, J. & Thomale, R.  Renormalization group analysis of competing quantum phases in the $J_{1}$-$J_{2}$ Heisenberg model on the kagome lattice.  _Phys. Rev. B_ 89, 020408 (2014).
  * (12) Jeschke, H. O., Salvat-Pujol, F., & Valentí, R.  First-principles determination of Heisenberg Hamiltonian parameters for the spin-1/2 kagome antiferromagnet ZnCu3(OH)6Cl2.  _Phys. Rev. B_ 88, 075106 (2013).
  * (13) Clissold, M. E., Leverett, P., Williams, P. A., Hibbs, D. E. & Nickel, E. H.  The structure of gillardite, the Ni-analogue of herbertsmithite, from Widgiemooltha, Western Australia.  _Can. Mineralog._ 45, 317 (2007).
  * (14) Colman, R. H., Sinclair, A. & Wills, A. S.  Magnetic and crystallographic studies of Mg-herbertsmithite, $\gamma$-Cu3Mg(OH)6Cl2 – a new $S=1/2$ kagome magnet and candidate spin liquid.  _Chem. Mater._ 23, 1811 (2011).
  * (15) Scott, J. D.  Crystal structure of a new mineral, söhngeite.  _The American Mineralogist_ 56, 355 (1971).
  * (16) Li, T., Wölfle, P. & Hirschfeld, P. J.  Spin-rotation-invariant slave-boson approach to the hubbard model.  _Phys. Rev. B_ 40, 6817–6821 (1989).
  * (17) Lechermann, F., Georges, A., Kotliar, G. & Parcollet, O.  Rotationally invariant slave-boson formalism and momentum dependence of the quasiparticle weight.  _Phys. Rev. B_ 76, 155102 (2007).
  * (18) In the charge-disordered state any electron has a 2/3 probability that a given neighbouring site is single-occupied and allows hopping, while in the ordered state half of the electrons are immobile, and the mobile ones have only 1/3 of the nearest neighbours single-occupied.
  * (19) Nagaoka, Y.  Ferromagnetism in a narrow, almost half-filled s band.  _Physical Review_ 147, 392 (1966).
  * (20) Hanisch, T., Uhrig, G. S. & Müller-Hartmann, E.  Lattice dependence of saturated ferromagnetism in the hubbard model.  _Phys. Rev. B_ 56, 13960–13982 (1997).
  * (21) Not only the coupling constant $\lambda$ is small in this case, the critical temperature $T_{c}\propto\exp(-1/\lambda^{2}),$ rather than $\exp(-1/\lambda)$.
  * (22) Harris, A. B., Kallin, C. & Berlinsky, A. J.  Possible Néel orderings of the kagome antiferromagnet.  _Phys. Rev. B_ 45, 2899–2919 (1992).
  * (23) Sigrist, M. & Ueda, K.  Phenomenological theory of unconventional superconductivity.  _Rev. Mod. Phys._ 63, 239–311 (1991).
  * (24) Platt, C., Hanke, W. & Thomale, R.  Functional renormalization group for multi-orbital fermi surface instabilities.  _Advances in Physics_ 62, 453 (2013).
  * (25) Nandkishore, R., Maciejko, J., Huse, D. A. & Sondhi, S. L.  Superconductivity of disordered dirac fermions.  _Phys. Rev. B_ 87, 174511 (2013).
  * (26) Kiesel, M. L., Platt, C., Hanke, W., Abanin, D. A. & Thomale, R.  Competing many-body instabilities and unconventional superconductivity in graphene.  _Phys. Rev. B_ 86, 020507 (2012).