Quantum phase transitions and dimensional reduction in antiferromagnets with interlayer frustration

For magnets with a fully frustrated interlayer interaction, we argue that the quantum phase transitions from a paramagnetic to an antiferromagnetic ground state, driven by pressure or magnetic field, are asymptotically three dimensional, due to interaction-generated nonfrustrated interlayer couplings. However, the relevant crossover scale is tiny, such that two-dimensional behavior occurs in an experimentally relevant low-temperature regime. In the pressure-driven case the phase transition may split, in which case an Ising symmetry related to interlayer bond order is broken before magnetism occurs. We discuss the relation of our results to recent experiments on $\mathrm{Ba}\mathrm{Cu}{\mathrm{Si}}_2{\mathrm{O}}_6$.

Figure 1

(Color online) (a) bct lattice of dimers. (b) Top view: two planes with sites shown as circles and crosses, and in-plane (interplane) coupling as solid (dashed) lines. (c) Illustration of the $Z_2$ symmetry breaking, where the diagonal interplane bonds develop a spontaneous asymmetry (see text), together with the simplest lattice compatible distortion. (d) $T=0$ phase diagrams for the coupled-dimer model (1), for the pressure-driven case with and without splitting (see text) and the field-driven case.

Figure 2

(Color online) Diagrams relevant for the order-parameter theory ${\mathcal{S}}_{\varphi }$. Lines are ${\varphi }_n\left({\vec{k}}_{\Vert }\right)$ propagators, the circle is the local four-point vertex $(\propto u_0)$, the cross is the interlayer hopping $(\propto \eta )$. (a) Unfrustrated vertical second-neighbor coupling—this is the ${\eta }^{\prime }$ term in (4), which is responsible for 3D behavior at lowest energies. (b) Interlayer density interaction, generating the $u_1$ and $u_2$ terms in (4). $u_1<0$ leads to collinear interplane spin correlations. (c) Vertical near-neighbor coupling in the ordered phase (the open square denotes the condensate).

Figure 3

(Color online) Triplon dispersion (solid) and boundary of the two-particle continuum (shaded), calculated using bond operators, for $J^{\prime }∕J=0.15$, $-J_z=J_{4z}=0.2J^{\prime }$, $H=0$. The inset shows an energy zoom into the almost flat vertical dispersion. The dominant effect of $H$ is a Zeeman shift of $-\alpha H$ of the triplons.