  The phase diagrams of highly frustrated quantum spin systems can exhibit
first-order quantum phase transitions and thermal critical points even in the
absence of any long-ranged magnetic order. However, all unbiased numerical
techniques for investigating frustrated quantum magnets face significant
challenges, and for generic quantum Monte Carlo methods the challenge is the
sign problem. Here we report on a general quantum Monte Carlo approach with a
loop-update scheme that operates in any basis, and we show that, with an
appropriate choice of basis, it allows us to study a frustrated model of
coupled spin-1/2 trimers: simulations of the trilayer Heisenberg
antiferromagnet in the spin-trimer basis are sign-problem-free when the
intertrimer couplings are fully frustrated. This model features a first-order
quantum phase transition, from which a line of first-order transitions emerges
at finite temperatures and terminates in a thermal critical point. The trimer
unit cell hosts an internal degree of freedom that can be controlled to induce
an extensive entropy jump at the quantum transition, which alters the shape of
the first-order line. We explore the consequences for the thermal properties in
the vicinity of the critical point, which include profound changes in the lines
of maxima defined by the specific heat. Our findings reveal trimer quantum
magnets as fundamental systems capturing in full the complex thermal physics of
the strongly frustrated regime.
