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Intermetallic compounds containing $f$-electron elements have beenprototypical materials for investigating strong electron correlations andquantum criticality (QC). Their heavy fermion ground state evoked by themagnetic $f$-electrons is susceptible to the onset of quantum phases, such asmagnetism or superconductivity, due to the enhanced effective mass ($m^{*}$)and a corresponding decrease of the Fermi temperature. However, the presence of$f$-electron valence fluctuations to a non-magnetic state is regarded ananathema to QC, as it usually generates a paramagnetic Fermi-liquid state withquasiparticles of moderate $m^{*}$. Such systems are typically isotropic, witha characteristic energy scale $T_0$ of the order of hundreds of kelvins thatrequire large magnetic fields or pressures to promote a valence or magneticinstability. Here we show that the intermediate valence compound$\alpha$-YbAlB$_4$ surprisingly exhibits both quantum critical behaviour and aLifshitz transition under low magnetic field, which is attributed to theanisotropy of the hybridization between the conduction and localized$f$-electrons. These findings suggest a new route to bypass the large valenceenergy scale in developing the QC.

Quantum critical behaviour and Lifshitz transition in intermediate valence $\alpha$-YbAlB$_4$