Quantum Spin Liquids Stabilized by Disorder in Non-Kramers Pyrochlores

TL;DR

This work addresses whether quantum spin liquids can be stabilized in non-Kramers pyrochlores by disorder. It models the system as a disordered transverse-field Ising model and solves it in real space using gauge mean-field theory (GMFT) with a parton representation, yielding a self-consistent quadratic spinon hopping problem whose spectrum is given by the eigenvalues of the mass matrix $M$. The main findings are that the quantum spin ice phase is robust against disorder up to the polarized transition, the Griffiths region near the transition is tiny, and an average description of disorder captures the behavior across most of the phase diagram; the spinon gap distributions remain smooth, and disorder can even delocalize spinons away from criticality. These results imply that disorder can stabilize quantum spin ice in real materials and provide guidance for experiments probing emergent photons and specific-heat signatures in the QSI regime.

Abstract

This study investigates the emergence of quantum spin liquid phases in pyrochlore oxides with non-Kramers ions, driven by structural randomness that effectively acts as a transverse field, introducing quantum fluctuations on top of the spin ice manifold. This is contrary to the naive expectation that disorder favors phases with short-range entanglement by adjusting the spins with their local environment. Given this unusual situation, it is essential to assess the stability of the spin-liquid phase with respect to the disorder. To perform this study, a minimal model for disordered quantum spin ice, the transverse-field Ising model, is analyzed using a formulation of gauge mean-field theory (GMFT) directly in real space. This approach allows the inclusion of disorder effects exactly and provides access to non-perturbative effects. The analysis shows that the quantum spin ice remains remarkably stable with respect to disorder up to the transition to the polarized phase at high fields, indicating that it can occur in real materials. Moreover, the Griffiths region of enhanced disorder-induced fluctuations appears tiny and restricted to the immediate vicinity of this transition due to the uniqueness of the low-energy excitations of the problem. For most of the phase diagram, an average description of the disorder captures the physical behavior well, indicating that the inhomogeneous quantum spin ice behaves closely to its homogeneous counterpart.

Figures (6)

• Figure 1: Phase diagram for the transverse-field Ising model on the pyrochlore lattice. The vertical axis shows the disorder fluctuations $\delta h/J$, and the horizontal axis shows the average value of the transverse field $\overline{h}$ divided by the critical value of the field $h_c$ that separates the QSI and the polarized phase in the clean case, $\delta h=0$. The color code indicates the average spinon gap, which serves as an order parameter. The values of $h_c$ are shown in Tab. \ref{['tab:dj-vs-hc']}. The results are for $L=8$.
• Figure 2: Gauge mean-field results for the random transverse field Ising model of Eq. \ref{['eq:h']} with a bimodal distribution of fields $h_i\in\{\overline{h}\pm\delta h\}$. The results are for a diamond lattice cluster with linear size $L=8$ ($N=1024$ sites). The black circles show the average constraint $\overline{\lambda}/J$ and the red squares denote the average effective local field as we vary the mean transverse field normalized by the disorder-dependent critical field $\overline{h_c}$, see Tab. \ref{['tab:dj-vs-hc']}. The solid lines are a fit to the data. The error bars, estimated from the data variance, are smaller than the symbol sizes. (a) $\delta h/J = 0.30$. Here we considered $3000$ realizations of disorder. (b) $\delta h/J = 0.70$. Here we considered $5000$ realizations of disorder.
• Figure 3: Gauge mean field statistics for the spinon gap $\omega_0$ on a diamond lattice cluster of linear size $L = 8$. We used $3000,~4000,~5000$ disorder realizations for $\delta h/J = 0.3, 0.5, 0.7$, respectively. (a) The average spinon gap $\overline{\omega}_0/J$ as we vary the mean transverse field normalized by the critical field $\overline{h_c}$ for $\delta h/J = 0.30$, black circles, and $\delta h/J = 0.70$, red circles. The error bars are smaller than the symbol sizes. The solid green line is the analytic result in the clean limit, $\omega_0/J\sim\sqrt{\lambda - h}$savary12a. (b) The spinon gap relative fluctuation for $\delta h/J = 0.20, 0.30, 0.40, 0.50, 0.70$. The solid brown is a fit of the maximum of the curves. (c) The spinon gap distributions at $\overline{h} = 0.50\overline{h_c}$ for $\delta h/J = 0.3, 0.5, 0.7$. (d) The same distribution as (c), but near the quantum critical point $\overline h = 0.99\overline{h_c}$.
• Figure 4: Results for the consecutive gap ratio statistics of the low-lying spinon modes $P(r)$. For each disorder realization, we use the five lowest eigenenergies of Eq. \ref{['eq:mf_Hamiltonian']} to compute $r$. We consider $5000$ disorder realizations. The solid red curve is the analytical result for the Poisson regime. The dashed black curve is the analytical result for the GOE case. (a)-(c) $P(r)$ for $\delta h/J = 0.7$ and $\overline{h}/\overline{h_c} = 0.15, 0.80, 0.99$. (d) Heat map of the average consecutive gap ratio for the low-lying energy levels, $\overline{r}$, as a function of $\delta h$ and $\overline{h}$. The blue and red shades indicate GOE and Poisson-like statistics, respectively.
• Figure 5: Effects of a finite $J_\pm$ on the gauge mean field results at disorder $\delta h/J = 0.70$ on a diamond lattice of linear size $L = 8$. (a)-(b) The consecutive gap ratio distribution computed from the lowest five energy levels at $\overline{h} = 0.15\overline{h_c}$ for $J_\pm/J = 0.00$ (a) and $J_\pm/J = 0.10$ (b). The red solid and black dashed lines are the analytical results for the Poisson and GOE statistics, respectively. (c) The average spinon gap as a function of the average value of the transverse field $\overline{h}$ normalized by the critical field $\overline{h_c}$. (d) Distribution of the spinon gap at $\overline{h} = 0.85\overline{h_c}$.