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low - temperature physics of correlated materials is often characterized by the competition between ordered phases and unconventional superconductivity .
tipically , a static mean - field description , implying negligible fluctuations beyond the limits of the ordered phase , is not valid in these systems .
nearly all dynamical probes show strong order parameter fluctuations , not only in the neighboring superconducting phases , which suggests a natural mechanism of pairing , but also in the strange metal , present at higher temperatures .
lithium purple bronze ( lipb ) , adds the ingredient of quasi - one - dimensionality to the problem and suggests the possibility that charge and spin fluctuations alone , without the existence of real order , might be responsible of superconductivity and anomalies of the normal phase .
the metallic phase of lipb , with chemical formula li@xmath0mo@xmath1o@xmath2 , has been characterized as a robust luttinger liquid ( ll ) in a series of angle resolved photoemission spectroscopy ( arpes ) experiments ranging different temperature regimes , sample growth techniques , photon energies , and data analysis procedures @xcite .
stm spectroscopy shows @xcite ll single - particle density of states and thermal and electric transport measurements are in complete disagreement with widemann - franz law @xcite .
when temperature is decreased , an upturn of the resistivity occurs at @xmath5 k @xcite and the material becomes superconducting at lower temperatures around @xmath6 k @xcite .    unlike other low - dimensional bronzes ,
the resistivity upturn of lipb @xcite is not associated with a lattice distortion ( see table 1 in ref . ) .
neither thermal expansion @xcite nor neutron scattering experiments @xcite have identified a phase transition at @xmath7 suggesting the idea of a soft crossover of electronic nature .
no gap has been clearly observed in the spectroscopies but optical conductivity measurements @xcite suggest the presence of a weak pseudogap .
recently , thermopower @xcite and nmr @xcite experiments have confirmed different aspects of the quasi - one - dimensionality of this material but the nature of the upturn remains a mistery .
the most recent study of superconducting properties @xcite confirms quantitatively that the large anisotropies observed in the upper critical field agree with those expected from the electrical resistivity in the metallic phase .
the coherence lengths perpendicular to the chains are larger than interchain distances and @xmath8 increases monotonically with decreasing temperature to values 5 times larger than the estimated paramagnetic pair - breaking field .
neither spin - orbit scattering nor strong - coupling superconductivity seem to explain this behavior suggesting the possibility of spin triplet superconductivity .
a quantitative comparison with experiments @xcite shows that superconductivity can be destroyed through orbital effects at fields higher than the clogston paramagnetic limit _ provided _ that the superconducting pairs are in the triplet state . + in the last years there has been a very important theoretical effort @xcite to reduce the complexity of the unit cell to microscopic hamiltonians reproducing different aspects of this phenomenology . in this article , we present a microscopic theory for the unconventional superconducting properties observed in li@xmath0mo@xmath1o@xmath2 .
based on a minimal extended hubbard model introduced in ref .
, we show that li@xmath0mo@xmath1o@xmath2 superconducts in the triplet channel when charge and spin fluctuations are enhanced , which may be also related with the upturn in resistivity at @xmath7 @xcite . using the random phase approximation ( rpa )
, we identify the cdw pattern characterized by two ordering wave vectors , @xmath3 and @xmath4 . in the proximity of those phases we evaluate and analize the superconducting vertex finding dominant p - wave triplet superconductivity with nodes on the fermi surface . within our methodology we find results compatible with the one presented in a very recent preprint @xcite
the electronic structure close to the @xmath9 and the quasi - one - dimensionality of the system derives from two parallel zig - zag mo - o chains per unit cell @xcite fig .
tight binding @xcite and dft @xcite band structure calculations agree that the mo - o orbitals of the chain give rise to four bands and two of them cross the fermi level .
arpes confirms the quasi - one - dimensionality of the fermi surface .
a slater - koster tight binding parametrization of the system has been propoposed in ref . and
the role of long - range coulomb couplings in the anomalies of the metallic phase has been also studied @xcite . here , we consider a strongly correlated model , which can capture the essential physics of li@xmath0mo@xmath1o@xmath2 @xcite consisting on an extended hubbard lattice with 4 mo - atoms per unit cell , which reads : @xmath10 where @xmath11 is the non - interacting tight - binding hamiltonian .
the one - electron hamiltonian can be expressed in terms of bloch waves with the following non - zero matrix elements , the intra - ladder : @xmath12 ev , and @xmath13 ev and the hoppings among chains : @xmath14 ev , as is shown in fig [ fig1 ] ( dotted cell ) .
the diagonalized hamiltonian : @xmath15 , leads to four bands denoted by @xmath16 , the two lowest ones cross the @xmath9 @xcite .
the fermi surface , close to one quarter - filling , @xmath17 , is shown in fig .
[ chi0 ] ( a ) .
the coulomb interaction terms in the hamiltonian includes on - site hubbard interaction ( @xmath18 ) , intra - ladder interaction with the following non - zero matrix elements : @xmath19 and @xmath20 and inter - ladder @xmath21 interactions : @xmath22 and @xmath23 , as is shown in fig [ fig1 ] ( dashed cell ) .
@xmath24    the interacting hamiltonian only includes density - density coulomb interaction contributions . within this work
, we have consider several combinations of parameters , all of them leading to essentially the same results presented here where we reduce the parameter space to two variables ( @xmath18 and @xmath25 ) .
we take the coulomb interaction among different sites with @xmath26 dependence , where @xmath27 is the distance among orbitals .
therefore , we parametrize the interactions by weighting the v s with the interatomic distances : @xmath28 .    [ cols="^,^ " , ]        values found for each @xmath18 . solid . the exponent in the luttinger density of states , @xmath29.,width=302 ]    to summarize ,
we have studied a microscopic extended hubbard model for lipb .
we have characterized the couplings promoting sc close to different charge ordering patterns . a detailed analyisis within the rpa approximation of the vertex shows triplet superconductivity with nodes on the fermi surface close to those ordered phases .
the relevance of these results is discussed in terms of the general experimental perspective of the material .
we thank j.w . allen , j. merino , l. taillefer for fruitful discussions .
we acknowledge financial support from mineco fis2012 - 37549-c05 - 03 .
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we find that in order to generate pairing eigenvalues of order 1 , we need to work within the rpa at interaction strengths quite close to the charge ( or spin ) instability .
however , graser et al .
@xcite mention that from the earlier experience comparing rpa with quantum monte carlo results on the single - band hubbard model @xcite expect that the bare interaction parameters appearing in the rpa expressions are renormalized , such that one is actually not so close to the true instability .
superconductivity in the quasi - one - dimensional material li@xmath0mo@xmath1o@xmath2 is analyzed based on a multiorbital extended hubbard model . 
 we found strong charge fluctuations at two different momenta @xmath3 and @xmath4 giving rise to two different charge ordered phases . 
 evaluating the superconducting vertex , we found superconductivity near strong charge fluctuations at @xmath3 . the order parameter has the p - wave symmetry with nodes on the fermi surface . 
 the metallic state displays a characteristic charge collective mode @xmath3 due to nesting and for on - site hubbard repulsion sufficiently large , a charge critical mode @xmath4 driven by coulomb repulsion , which softens at the proximity to the transition . 
 the results are quite robust for different coupling parametrizations . 
 a phase diagram discussing the relevance of the model to the physics of the material is proposed .