Understanding quantum critical magnetism in Han purple

Han purple, a pigment used to paint the famous terracotta army (left), consists of stacked two-dimensional bilayers hosting Cu spin dimers (centre). The ground state of this system is a “quantum disordered’’ state, in which every dimer hosts a spin singlet. In a magnetic field, this state persists up to a critical field value where the system undergoes a quantum phase transition (QPT) into a magnetically ordered phase that may be considered as a condensate of bosonic quasiparticles.

Magnetisation, magnetic torque and nuclear magnetic resonance (NMR) measurements were interpreted as showing that the system turns from three-dimensional (3D) to two-dimensional (2D) on approaching the QPT from the ordered side. This result contrasts starkly with the expected universal 3D physics in the low-energy, long-wavelength limit when the correlation length diverges in the quantum critical regime around the QPT.

Multiple mechanisms proposed to explain this phenomenon assumed geometrical frustration of the interactions between the bilayers (inter-bilayer), which emerges naturally for antiferromagnetic interactions between dimer units within the bilayers (intra-bilayer). However, an ab inito study proposed that these intra-bilayer interactions are effectively ferromagnetic, precluding any inter-bilayer frustration and thus reopening the search for a mechanism underlying the mysterious critical behaviour.

Researchers at EPFL, PSI and the University of Toulouse have performed high-resolution neutron spectroscopy measurements across multiple Brillouin zones of the system at zero applied field to determine all of the magnetic interactions. Their data revealed effectively ferromagnetic intra-bilayer interactions, in full agreement with the ab initio study, and demonstrated that the crystal structure contains three magnetically inequivalent bilayer types, A, B, and C in ratios 3:2:1 (centre, right top).

By state-of-the-art quantum Monte Carlo simulations, the research team modelled the phase boundary and bilayer quasiparticle occupation around the field-induced QPT (right bottom), finding excellent agreement with the earlier magnetic measurements. These simulations proved that the modulated magnetic structure, with its three bilayer types, introduces a new energy scale in the problem and hence a regime of anomalous critical scaling above the QPT. This had provided the appearance of dimensional reduction and depressed the true 3D quantum critical regime to a very small region around the transition.

The technological capabilities for building layered magnetic systems from atomically thin materials are maturing rapidly. Thus the discovery that structural modulation can create unconventional effective scaling may have broad applicability in designing the physical properties of quantum engineered heterostructures.