Quantum Spin Ice with Frustrated Transverse Exchange: From a $\pi$-Flux Phase to a Nematic Quantum Spin Liquid

Quantum spin ice materials, pyrochlore magnets with competing Ising and transverse exchange interactions, have been widely discussed as candidates for a quantum spin-liquid ground state. Here, motivated by quantum chemical calculations for Pr pyrochlores, we present the results of a study for frustrated transverse exchange. Using a combination of variational calculations, exact diagonalization, numerical linked-cluster and series expansions, we find that the previously studied $U(1)$ quantum spin liquid, in its $\pi$-flux phase, transforms into a nematic quantum spin liquid at a high–symmetry, $SU(2)$ point.

Figure 1

Phase diagram of the quantum spin ice model ${\mathcal{H}}_{XXZ}$ [Eqs. (1) and (2)]: (a) Quantum phase diagram found in cluster-variational calculations for $T=0$. Quantum spin liquids descended from spin ice, ${\mathrm{QSI}}_0$ and ${\mathrm{QSI}}_{\pi }$, compete with easy-plane antiferromagnetic order (${\mathrm{AF}}_{\perp }$); all-in all-out order (AIAO); and a nematic QSL (${\mathrm{QSN}}_{\perp }$). ${\mathrm{QSI}}_{\pi }$ and ${\mathrm{QSN}}_{\perp }$ are connected through an $SU(2)$-symmetric point. (b) Phase diagram found in classical Monte Carlo simulations for $T>0$ (cf. Ref. [25]). Three spin liquids; spin ice (SI); the easy-plane spin liquid (${\mathrm{SL}}_{\perp }$); and a pseudo-Heisenberg antiferromagnet (PHAF), compete with a nematic spin liquid (${\mathrm{SN}}_{\perp }$), and ${\mathrm{AF}}_{\perp }$ and AIAO order. An additional disordered regime (${\mathrm{SL}}^{\prime }$) shares the correlations of ${\mathrm{SL}}_{\perp }$ and AIAO. The white circles indicate the radial, logarithmic, temperature scale. Simulation details are given in the Supplemental Materials [30].

Figure 2

Condensation of two-magnon bound states within the AIAO phase, indicating the onset of spin-nematic order. Gaps to the lowest one-magnon and two-magnon excitations of the AIAO phase, $\mathrm{\Delta }(\delta S^z=1)$ and $\mathrm{\Delta }(\delta S^z=2)$, are shown as a function of the Hamiltonian parameter $\theta$ [Eq. (2)]. For $\delta S^z=1$, the gap is exact, while for $\delta S^z=2$, it is estimated numerically for a 128 site cluster. As $\theta$ decreases towards $\theta =0.612\pi$ the two-magnon state comes below the one magnon state and then condenses. The condensation of two-magnon bound states indicates incipient nematicity [28]. The resultant phase boundary between ${\mathrm{QSN}}_{\perp }$ and AIAO, $\theta \approx 0.612\pi$, is close to that found in CVAR $\theta \approx 0.613\pi$ [cf. Fig. 1], shown as a dashed line.

Figure 3

Finite-temperature properties of frustrated quantum spin ice [Eq. (1)]. (a) Susceptibility ${\chi }_{\mathrm{nem}}(T)$ associated with spin-nematic order [Eq. (6)], from HTE. Temperature in (a) is shown in units of the Ising exchange $J_{zz}=J\cos (\theta )$. Different curves for a given value of $\theta$ correspond to different Padé approximants. For $\theta >(\pi /4)$, ${\chi }_{\mathrm{nem}}(T)$ shows an upturn at low temperatures, consistent with an approach to spin-nematic order. (b) Heat capacity $C$, as a function of reduced temperature $T/T_m$, calculated within NLCE. Here, $T_m$ is the temperature of the heat capacity maximum; different curves for the same $\theta$ represent different orders of NLCE; agreement between these indicates convergence. Plots of $C(T/T_m)$ collapse onto one another for $\theta \gtrsim 0.17\pi$, consistent with an extended regime where finite-temperature properties are controlled by the zero-temperature $SU(2)$ point $\theta =(\pi /4)$, reminiscent of quantum criticality.