Spin liquid and quantum phase transition without symmetry breaking in a frustrated three-dimensional Ising model

We show that the highly frustrated transverse-field Ising model on the three-dimensional pyrochlore lattice realizes a first-order phase transition without symmetry breaking between the low-field Coulomb quantum spin liquid and the high-field polarized phase. The quantum phase transition is located quantitatively by comparing low- and high-field series expansions. Furthermore, the intriguing properties of the elementary excitations in the polarized phase are investigated. We argue that this model can be achieved experimentally by applying mechanical strain to a classical spin-ice material composed of non-Kramers spins such as ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. Taken together with our results, this provides an experimental platform to study quantum spin liquid physics.

Figure 1

Phase diagram of the TFIM on the pyrochlore lattice as a function of $h/J$ consisting of the low-field CQSL and the high-field polarized phase separated by a first-order phase transition at $h_{\mathrm{c}}/J_{\mathrm{c}}\approx 0.6$ shown as a black filled square. Four-site unit cells of the pyrochlore lattice are shown as dark tetrahedra. Left illustration: Dark arrows denote the three unit cell vectors. Green hexagon exemplifies a hexagon on which the ring-exchange $K$ term in Eq. (6) acts on alternating spins. Right illustration: Moving elementary spin-flip excitation above the high-field polarized state.

Figure 2

Illustration of the four $1qp$ bands ${\omega }_n\left(\vec{k}\right)$ with $n\in \{1,2,3,4\}$ in the polarized phase for $x=0.1$ along a characteristic path in the three-dimensional Brillouin zone using the bare order-11 series expansion. The lowest band is twofold degenerate for all momenta except for $\vec{k}=(0,0,0)$ where the degeneracy is three. Inset: Zoom on the two low-energy bands.

Figure 3

High-field gap $\mathrm{\Delta }/h$ as a function of $u=x/(x+1)$ using the bare order-11 series of the high-field expansion. The bare series of order 10 and 11 are shown as dashed lines. Other lines refer to various DlogPadé extrapolants $[n,m]$.

Figure 4

Comparison of the ground-state energies ${\epsilon }_0$ from the low- and high-field expansions as a function of $\phi$ setting $J=cos\phi$ and $h=sin\phi$. Dashed vertical line refers to the mean-field result from Ref. [2] and solid vertical line corresponds to location of the first-order quantum phase transition as obtained from the crossing of the series expansions.