Crossovers and critical scaling in the one-dimensional transverse-field Ising model*

We consider the scaling behavior of thermodynamic quantities in the one-dimensional transverse field
Ising model near its quantum critical point (QCP). We find that the crossovers obey a general scaling ansatz, and so does the critical scaling behavior of the specific heat and magnetic expansion coefficient. Furthermore, the Grüneisen ratio diverges in a power-law way when the QCP is accessed as a function of the transverse field at zero temperature. However, at the critical field, upon decreasing the temperature, the Grüneisen ratio approaches a constant instead of showing the expected divergence. This unusual result can be understood in terms of a peculiar form of the quantum critical scaling function for the free energy; the contribution to the Grüneisen ratio vanishes at the linear order in a suitable Taylor expansion of the scaling function. Our results establish the telltale thermodynamic signature of a transverse-field Ising chain, and will thus facilitate the experimental identification of this model quantum-critical system in real materials.