Cu(pz)2(ClO4)2 (with pz denoting pyrazine, C4H4N2) is among the best realizations of a two-dimensional spin-1/2 square-lattice antiferromagnet. Below T_N = 4.21 K, its weak interlayer couplings induce a 3D magnetic order, strongly influenced by external magnetic fields and/or hydrostatic pressure. Previous work, focusing on the [H, T] phase diagram, identified a spin-flop transition, resulting in a field-tunable bicritical point. However, the influence of external pressure has not been investigated yet. Here we explore the extended [p, H, T] phase diagram of Cu(pz)2(ClO4)2 under pressures up to 12 kbar and magnetic fields up to 7.1 T, via magnetometry and 35Cl nuclear magnetic resonance (NMR) measurements. The application of magnetic fields enhances T_XY , the crossover temperature from the Heisenberg to the XY model, thus pointing to an enhancement of the effective anisotropy. The applied pressure has an opposite effect [dT_N/dp = 0.050(8) K/kbar], as it modifies marginally the interlayer couplings, but likely changes more significantly the orbital reorientation and the square-lattice deformation. This results in a remodeling of the effective Hamiltonian, whereby the field and pressure effects compensate each other. Finally, by comparing the experimental data with numerical simulations we estimate T_BKT, the temperature of the Berezinskii-Kosterlitz-Thouless topological transition and argue why it is inaccessible in our case.
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