Finite-temperature dynamics in 0-flux and $\pi$-flux quantum spin ice: Self-consistent exclusive boson approach

Quantum spin ice (QSI) is an emblematic three-dimensional $U\left(1\right)$ quantum spin liquid (QSL) on the pyrochlore lattice that hosts gapless photon-like modes and spinon excitations. Despite its notable status and the current rise of strong material candidates ${\mathrm{Ce}}_2$(Zr, Sn, ${\mathrm{Hf})}_2{\mathrm{O}}_7$, there are still only a few analytical approaches to model the low-energy physics of QSI. These analytical methods are essential to gain insight into the physical interpretation of measurements. We here introduce the self-consistent exclusive boson representation (SCEBR) to model emergent spinon excitations in QSI. By treating the presence of other emergent charges in an average way, the SCEBR extends the range of validity of the exclusive boson representation previously introduced in [Hao, Day, and Gingras, Phys. Rev. B 90, 214430 (2014)] to numerous cases of physical relevance. We extensively benchmark the approach and provide detailed analytical expressions for the spinon dispersion, the Bogoliubov transformation that diagonalizes the system, and the dynamical spin structure factor for 0- and $\pi$-flux QSI. Finite-temperature properties are further investigated to highlight essential differences between the thermodynamic behavior of the 0- and $\pi$-flux phases. We notably show that the SCEBR predicts a reduction of the spinon bandwidth with increasing temperature, consistent with previous quantum Monte Carlo results, through suppression of spinon hopping by thermal occupation. The SCEBR thus provides a powerful analytical tool to interpret experiments on current and future candidate material that has several advantages over other widely used methods.

Figure 1

(a) The pyrochlore lattice formed by a network of up- (green) and down-pointing (purple) tetrahedra. (b) The pyrochlore lattice sites sit at the bond center of its parent (i.e., premedial) diamond lattice. (c) Ground-state energy per diamond lattice unit cell as a function of transverse coupling in the low-density approximation to the exclusive boson formulation for 0- and $\pi$-flux QSI.

Figure 2

(a) Average tetrahedron occupation and (b) spinon gap as a function $J_{\pm }/J_{\parallel }$ using the low-density exclusive boson approach (dashed lines) and the self-consistent exclusive boson representation (full lines). Calculations use 0-flux and $\pi$-flux QSI for $J_{\pm }>0$ and $J_{\pm }<0$, respectively (i.e., the lowest-energy state). The black dash-dotted line in panel (a) is the average occupation in the infinite-temperature limit $\langle n\rangle =3/4$.

Figure 3

(a) Spinon dispersion and corresponding single spinon density of states ${\rho }^{\left(1\right)}\left(\omega \right)$, and (b) dynamical spin structure factor $S^{zz}(\mathbf{q},\omega )$ (assuming $J_{\parallel }=J_{xx}$ or $J_{\parallel }=J_{yy}$) and two-spinon density of states ${\rho }^{\left(2\right)}\left(\omega \right)$ for 0-flux QSI with $J_{\pm }/J_{\parallel }=0.06$. White lines represent the lower and upper edges of the two-spinon continuum.

Figure 4

(a) Spinon dispersion and corresponding single spinon density of states ${\rho }^{\left(1\right)}\left(\omega \right)$, and (b) dynamical spin structure factor $S^{zz}(\mathbf{q},\omega )$ (assuming $J_{\parallel }=J_{xx}$ or $J_{\parallel }=J_{yy}$) and two-spinon density of states ${\rho }^{\left(2\right)}\left(\omega \right)$ for $\pi$-flux QSI with $J_{\pm }/J_{\parallel }=-0.2$. White lines represent the lower and upper edges of the two-spinon continuum.

Figure 5

(1) Equal-time structure factor ${\mathcal{S}}^{zz}\left(\mathbf{q}\right)$ in the $(h,h,l)$ plane and its contribution to the (2) spin-flip and (3) nonspin flip channel with neutrons polarized perpendicular to the scattering plane for (a) $\pi$-flux QSI with $J_{\pm }/J_{\parallel }=-0.2$ and (b) 0-flux QSI with $J_{\pm }/J_{\parallel }=0.06$.

Figure 6

(a) Average tetrahedron occupation as a function of temperature and transverse coupling. Also shown is the approximate scaling of the vison gap ${\mathrm{\Delta }}_{\text{vison}}\sim 12{\left|J_{\pm }\right|}^3/J_{\parallel }^2$ and a contour line for the tetrahedron occupation in the $T\to \infty$ limit $\langle n\rangle =3/4$. (b) The associated renormalized transverse coupling ${\tilde{J}}_{\pm }=J_{\pm }/(1+\langle n\rangle )$.

Figure 7

Finite-temperature dynamical spin structure factor $S^{zz}(\mathbf{q},\omega )$ (assuming $J_{\parallel }=J_{xx}$ or $J_{\parallel }=J_{yy}$) for 0-QSI with $J_{\pm }/J_{\parallel }=0.06$ at (a) $k_{B}T/J_{\parallel }=0.1$, (b) $k_{B}T/J_{\parallel }=0.2$, (c) $k_{B}T/J_{\parallel }=0.3$, and (d) $k_{B}T/J_{\parallel }=0.4$. The multiplicative factor is the scale of the colorbar with respect to results at $k_{B}T/J_{\parallel }=0.1$.

Figure 8

Finite-temperature dynamical spin structure factor $S^{zz}(\mathbf{q},\omega )$ (assuming $J_{\parallel }=J_{xx}$ or $J_{\parallel }=J_{yy}$) for $\pi$-QSI with $J_{\pm }/J_{\parallel }=-0.2$ at (a) $k_{B}T/J_{\parallel }=0.1$, (b) $k_{B}T/J_{\parallel }=0.2$, (c) $k_{B}T/J_{\parallel }=0.3$, and (d) $k_{B}T/J_{\parallel }=0.4$. The multiplicative factor is the scale of the colorbar with respect to results at $k_{B}T/J_{\parallel }=0.1$.