The success of Fermi-liquid (FL) theory in describing the properties of most ordinary three-dimensional metals makes it one of the triumphs of twentieth-century theoretical physics. Its wide-ranging applicability is testament to the validity of describing a system of interacting electrons by mapping its low-lying quasiparticle excitations onto a Fermi gas of non-interacting electrons. Perhaps the most striking realization of this one-to-one correspondence is the validity of the Wiedemann–Franz (WF) law in almost all known theoretical1 2 3 4 and experimental5 6 7 cases. The WF law states that the ratio of the electronic thermal conductivity κ e to the electrical conductivity σ at a given temperature T is equal to a constant called the Lorenz number or Lorenz ratio, L 0=κ e/σT=(π 2/3)(k B/e)2 and reflects the fact that thermal and electrical currents are carried by the same fermionic quasiparticles. Although the WF law is most applicable in the zero temperature (impurity scattering) limit, it is found to hold equally well at room temperature once all inelastic scattering processes become active8. A marked deviation from the WF law is theoretically predicted when electrons are spatially confined to a single dimension. In systems that are strictly one-dimensional (1D), even weak interactions destroy the single particle FL picture in favour of an exotic Tomonaga–Luttinger liquid (TLL) state in which the fundamental excitations are independent collective modes of spin and charge, referred to, respectively, as spinons and holons. As heat is transported by entropy (spin and charge) and electric current by charge alone, spin–charge separation is a viable mechanism for the violation of the WF law9 10 11 12. Physically, repulsive interactions in a disordered 1D chain can inhibit the propagation of holons relative to that of spinons, leading to a strongly renormalized Lorenz number9. Experimental signatures of TLL physics have been seen in the spectral response of a number of 1D structures13 14 15 and bulk crystalline solids16 17 18 19 20. Although the ratio κ e/σT can in principle provide a direct means of distinguishing between FL and TLL states at low energies, there have been no confirmed reports to date of WF law violation in any 1D conductor. Identifying such systems, particularly bulk systems, is important as it might then allow one to tune, chemically or otherwise, the effective interchain coupling and thereby drive the system from one electronic state to the other. This would then open up the possibility of exploring the TLL-to-FL crossover and the nature of the excitations in the crossover regime. Here we report a study of the electrical and thermal conductivity tensors of the purple bronze Li0.9Mo6O17, a quasi-1D conductor whose (surface-derived) photoemission lineshapes19, and density of states profiles20 contain features consistent with TLL theory. Results Electrical resistivity of Li0.9Mo6O17 As shown in Figure 1a, Li0.9Mo6O17 possesses a set of weakly coupled zigzag chains of MoO6 octahedra with a hole-concentration, believed to be close to half-filling21, running parallel to the crystallographic b axis. The T-dependence of the b axis resistivity, plotted in Figure 1B, varies linearly with temperature above 100 K, then as T is lowered, ρ b(T) becomes superlinear. Below around 20 K, Li0.9Mo6O17 undergoes a crossover from metallic to insulating-like behaviour, ascribed to the formation of a putative charge density wave22 23. Also plotted in Figure 1b is the interchain resistivity ρ a(T). The anisotropy in the electrical resistivity, ρ a∼100ρ b ( / whereas L xy /L 0∼ 2/ 2, where is the Fermi-surface averaged mean free path for entropy (charge) transport, respectively. Hence, in ordinary metals, L xy /L 0 is expected to vary as (L xx /L 0)2, as found experimentally25. In Li0.9Mo6O17, however, (L xx /L 0)2 does not scale with L xy /L 0 and cannot be made to scale with L xy /L 0 for any reasonable estimate for the phonon contribution to κ xx (κ b). This, together with the unprecedented enhancement of the WF ratio by several orders of magnitude, provides compelling evidence for the breakdown of the conventional FL picture in this quasi-1D conductor. Discussion Before discussing the violation of the WF law in Li0. 9Mo6O17 in terms of 1D correlation physics, we first consider alternative scenarios based on localization effects. In non-interacting systems, it has been shown theoretically that the WF law is robust to impurity scattering of arbitrary strength up to the Anderson transition1 2 3 4. In strongly correlated electron systems, however, the opening of a Mott gap can lead to a strong reduction of the electrical conductivity whereas the transport of heat, through spin fluctuations, can remain high10. Localization corrections associated with electron–electron interactions are also believed to induce corrections to κ e that do not scale with the WF ratio, leading to an enhancement in L/L 0 (ref. 28). Such interaction corrections only appear in the diffusive limit, however, below a characteristic energy scale k B T d =ћ/2πτ determined by the impurity scattering rate 1/τ. Estimates for 1/τ in Li0.9Mo6O17 from in-chain resistivity or magnetoresistance measurements give T d values of order several tens of Kelvin. Whereas this estimate for the diffusion limit is consistent with the temperature (T min∼20 K) below which the resistivity starts to increase with decreasing T, the behaviour of the magnetoresistance below T min is found to be more consistent with density wave formation than localization corrections23. In addition, the strong violation of the WF law is observed in the metallic regime between T min and room temperature, and more significantly, above 100 K where the resistivity itself is strictly T-linear. Collectively, these observations appear to rule out localization as the origin of the WF law violation in Li0.9Mo6O17. Turning now to the issue of dimensionality, the form of the enhancement of the WF ratio in Li0.9Mo6O17 is at least qualitatively consistent with the original theoretical prediction for a spinless TLL9. An enhancement in L xx /L 0 originates from the fact that while heat can be transmitted through a non-magnetic impurity (via spinons), the latter acts as a near-perfect reflector (back-scatterer) of charge (holons). According to this picture, L xx is predicted to be of order L 0/K at high T, where K is the dimensionless conductance or Luttinger parameter, and as temperature is lowered, L xx /L 0 varies as a power-law, L xx /L 0∼T 4−2/K which diverges for K 1 mΩ cm) ρ b values have been reported, ρ b (T) tends to show a sub-linear T-dependence below 300 K. To isolate the in-chain resistivity extreme care is needed to electrically short out the sample in the two directions perpendicular to the chain and thus ensure that current flow between the voltage contacts is uniaxial. In our experiments, this is achieved either by coating conductive paint or evaporating gold strips across the entire width of the sample in the two orthogonal current directions. The mounting configuration is shown as an inset in Supplementary Figure S2. The zero field measurements were carried out for 4.2 K
