Continuous and discontinuous quantum phase transitions in a model two-dimensional magnet

Abstract

The Shasty-Sutherland model, which consists of a set of spin 1∕2 dimers on a 2D square lattice, is simple and soluble but captures a central theme of condensed matter physics by sitting precariously on the quantum edge between isolated, gapped excitations and collective, ordered ground states. We compress the model ShastrySutherland material, SrCu 2 ðBO 3 Þ 2 , in a diamond anvil cell at cryogenic temperatures to continuously tune the coupling energies and induce changes in state. High-resolution X-ray measurements exploit what emerges as a remarkably strong spin-lattice coupling to both monitor the magnetic behavior and the absence or presence of structural discontinuities. In the low-pressure spin-singlet regime, the onset of magnetism results in an expansion of the lattice with decreasing temperature, which permits a determination of the pressure-dependent energy gap and the almost isotropic spin-lattice coupling energies. The singlet-triplet gap energy is suppressed continuously with increasing pressure, vanishing completely by 2 GPa. This continuous quantum phase transition is followed by a structural distortion at higher pressure. O ne of the first real-world examples encountered in elementary quantum mechanics is the hydrogen atom, where two correlated spin 1∕2 particles, the electron and the proton, combine in multiple configurations to define a set of discrete energy levels. The ground state singlet superposes up and down configurations of the spins to yield a state with total spin zero; the excited state triplet provides three ways to produce a total spin of one. The fundamental quantum mechanics of spin singlets physically placed on ladders or constrained to sheets is a natural extension, with a rich set of possibilities for collective, interacting states from spin liquids to magnets to exotic superconductors (1-4). A particularly important topology involves spin 1∕2 orthogonal dimers on a square lattice, the so-called Shastry-Sutherland model (5), because it is exactly soluble. Here we exploit a remarkably strong coupling of the spin degrees of freedom to the lattice in a model Shastry-Sutherland system to probe the evolution of the collective spin configurations with temperature and pressure using X-rays and diamond anvil cell technology. Thermal excitations from singlet to triplet define a gap, which disappears as pressure P tunes the ratio between inter-and intradimer coupling energies. This continuous quantum phase transition occurs at P ¼ 2 GPa, followed by a first-order transition at P approximately 4.5 GPa, which we associate with the full-fledged onset of long-range antiferromagnetic order.Insulating SrCu 2 ðBO 3 Þ 2 (SCBO) successfully captures the physics of the Shastry-Sutherland model (6), with corner-sharing Cu 2þ S ¼ 1∕2 dimers lying on a square lattice and only weak interactions between adjacent planes. Under ambient conditions, the system has tetragonal symmetry with a ¼ 8.995 Å and c ¼ 6.649 Å. ESR (7) and neutron (8) data establish that the spin interactions in SCBO are well describ...

Emergence of long-range order in sheets of magnetic dimers

“…1, which has been eminently successful in describing the magnetic behavior of SCBO at low pressure and low temperature (7). If the magnetic structure of SCBO were completely described as a topological realization of the pure 2D Shastry-Sutherland (Fig.…”

“…Above the monoclinic transition at P ∼ 4.5 GPa, the presence or absence of diffraction intensity at these locations serves to indicate which symmetries associated with the tetragonal I-42m parent structure are preserved in the reconfigured lattice. We performed high-resolution single-crystal neutron diffraction at P = 5.5 ± 0.2 GPa, well above the 4.5-GPa pressure for which SCBO has been predicted to eventually enter an antiferromagnetic ordered phase (7,8). We plot in Fig.…”

“…Hydrostatic pressure can be used to tune x, with a transition into an intermediate phase observed at P ∼2 GPa (7,8). The magnetic characteristics of the singlet-triplet transition emerge even in X-ray scattering experiments due to a remarkably robust spin-lattice coupling (7,9,10). At P ∼4.5 GPa a second phase transition has been observed, marked by a structural transition from tetragonal to monoclinic ordering (7,11).…”

“…SCBO has a nonmagnetic, singlet ground state (5), with 0.6 < x < 0.69 (6). Hydrostatic pressure can be used to tune x, with a transition into an intermediate phase observed at P ∼2 GPa (7,8). The magnetic characteristics of the singlet-triplet transition emerge even in X-ray scattering experiments due to a remarkably robust spin-lattice coupling (7,9,10).…”

“…The magnetic characteristics of the singlet-triplet transition emerge even in X-ray scattering experiments due to a remarkably robust spin-lattice coupling (7,9,10). At P ∼4.5 GPa a second phase transition has been observed, marked by a structural transition from tetragonal to monoclinic ordering (7,11). In this article, we use both X-ray and neutron scattering measurements to illuminate the structural and magnetic configurations that permit long-range order to emerge for P > 5 GPa, directly identifying for the first time, to our knowledge, the predicted antiferromagnetic state.…”