Quantum spin ice on the breathing pyrochlore lattice

The Coulombic quantum spin liquid in quantum spin ice is an exotic quantum phase of matter that emerges on the pyrochlore lattice and is currently actively searched for. Motivated by recent experiments on the Yb-based breathing pyrochlore material ${\mathrm{Ba}}_3{\mathrm{Yb}}_2{\mathrm{Zn}}_5{\mathrm{O}}_{11}$, we theoretically study the phase diagram and magnetic properties of the relevant spin model. The latter takes the form of a quantum spin ice Hamiltonian on a breathing pyrochlore lattice, and we analyze the stability of the quantum spin liquid phase in the absence of the inversion symmetry which the lattice breaks explicitly at lattice sites. Using a gauge mean-field approach, we show that the quantum spin liquid occupies a finite region in parameter space. Moreover, there exists a direct quantum phase transition between the quantum spin liquid phase and featureless paramagnets, even though none of theses phases break any symmetry. At nonzero temperature, we show that breathing pyrochlores provide a much broader finite-temperature spin liquid regime than their regular counterparts. We discuss the implications of the results for current experiments and make predictions for future experiments on breathing pyrochlores.

Figure 1

(a) The breathing pyrochlore lattice. The letter “u” (“d”) refers to the up- (down-)pointing tetrahedra. The bond lengths on up and down tetrahedra are different. (b) The phase diagram of the Hamiltonian $H$ in Eq. (1) at $T=0$. See the main text for a detailed discussion.

Figure 2

Phase diagram in the $\kappa \text{−}{J}_{\pm }/J_{zz}\text{−}T/J_{zz}$ space. The three-dimensional surface indicates the first-order transition between the phases akin to the zero-temperature phases, namely the QSL, AFM, ${\mathrm{PM}}_{\mathrm{u},\mathrm{d}}$ (below the surface), and TSL (above the surface). The colors on the surface correspond to those of the phases immediately below the surface.

Figure 3

The ring exchange as a function of $\kappa$. The (red) dot is the position of the minimum and is at $\kappa \approx 0.92$.