New phases of matter emerge at the edge of magnetic instabilities. In local moment systems, such as heavy fermions, the magnetism can be destabilized by pressure, chemical doping, and, rarely, by magnetic field, towards a zero-temperature transition at a quantum critical point (QCP). Even more rare are instances of QCPs induced by pressure or doping in itinerant moment systems, with no known examples of analogous field-induced textit{T} = 0 transitions. Here we report the discovery of a new itinerant antiferromagnet with no magnetic constituents, in single crystals of Ti$_3$Cu$_4$ with $T_N$ = 11.3 K. Band structure calculations point to an orbital-selective, spin density wave ground state, a consequence of the square net structural motif in Ti$_3$Cu$_4$. A small magnetic field, $H_C$ = 4.87 T, suppresses the long-range order via a continuous second-order transition, resulting in a field-induced QCP. The magnetic Gruneisen ratio diverges as $H rightarrow H_C$ and $Trightarrow0$, with a sign change at $H_C$ and $T^{-1}$ scaling at $H~=~H_C$, providing evidence from thermodynamic measurements for quantum criticality for $H parallel c$. Non-Fermi liquid (NFL) to Fermi liquid (FL) crossover is observed close to the QCP, as revealed by the power law behavior of the electrical resistivity.
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