## Fifth-order susceptibility unveils the growth of thermodynamic amorphous order in glass-formers

Glasses are ubiquitous in daily life and technology. However the microscopic mechanisms generating this state of matter are still controversially debated : glasses are considered either as merely hyper-viscous liquids or as resulting from a genuine thermodynamic phase transition towards a rigid state. We show that third and fifth order susceptibilities provide a definite answer to this longstanding controversy. Indeed, if an amorphous thermodynamic order is growing upon cooling, the fifth order response should be more anomalous in temperature and in frequency than the third order one. This is because each of the nonlinear responses is related to the size of amorphously ordered domains raised to some exponent ; but the higher the order of the response, the larger the exponent.

Owing to the extreme smallness of nonlinear responses, we have performed high precision dielectric experiments in two independent setups, yielding consistent results. More precisely, we have measured the third and fifth-order response respectively at the third and fifth harmonics onto supercooled glycerol and propylene carbonate.

We find strong support for theories based upon thermodynamic amorphous order. Moreover, when lowering temperature, we find that the growing transient domains are compact, i.e. their fractal dimension df=3. The glass transition may thus represent a new class of critical phenomena, different from canonical second-order phase transitions for which df<3.