Stability of algebraic spin liquids coupled to quantum phonons

TL;DR

We examine the stability of the U(1) Dirac spin liquid (DSL) on the triangular lattice $J_1$-$J_2$ Heisenberg model against spin-phonon coupling. Using variational Monte Carlo with both static distortions and fully dynamical phonons, guided by the QED$_3$ continuum, we extract the monopole scaling dimension $\Delta_\Phi$ and demonstrate a monopole-driven instability toward a valence-bond solid (VBS) order. Dimer-dimer correlations decay algebraically as ${\mathcal C}(\mathbf{r}) \propto r^{-2\Delta_\Phi}$ with $\Delta_\Phi \approx 1.23$–$1.25$, and the spin-Peierls transition appears at a finite critical spin-lattice coupling that depends on the phonon frequency $\omega$, with a characteristic 12-site VBS pattern emerging in the static limit and a similar instability in the dynamical case. The results yield a phase diagram where the DSL remains stable at small $g$ but yields to a DSL-to-VBS transition for larger couplings, and they establish a quantitative link between monopole dynamics in QED$_3$ and lattice couplings in two dimensions. These findings provide concrete predictions for experimental signatures in candidate materials and elucidate the interplay between algebraic spin liquids and lattice degrees of freedom in 2D.

Abstract

Algebraic spin liquids are quantum disordered phases of insulating magnets which exhibit fractionalized gapless excitations and power-law correlations. Quantum spin liquids in this category include the experimentally established 1D Luttinger liquid, as well as the U(1) Dirac spin liquid (DSL) which has been a focus of recent candidate materials searches. Most notably, several exchange-frustrated Heisenberg materials on the triangular lattice have shown evidence of the U(1) DSL. In this work, we measure the algebraic correlations of spin-singlet excitations in the $J_1$-$J_2$ antiferromagnetic Heisenberg model on the triangular lattice, prompting a detailed investigation of this model's stability under spin-phonon coupling using variational Monte Carlo. As seen before in 1D spin chains, we observe a low-temperature transition from a U(1) DSL to valence bond order and predict the parameter regime where the model realizes a stable DSL ground state. To achieve this, we employ a series of finite-size scaling Ansätze inspired by the low-energy DSL's conformal description in terms of quantum electrodynamics, and show that emergent monopole operators drive the instability. We compare the physics of this transition to the 1D Luttinger liquid throughout our analysis. We derive the regime of stability against spin-Peierls ordering and argue that the DSL ground state might still be achievable in candidate materials, despite its tendency to valence bond solid ordering.

Figures (5)

• Figure 1: Phase diagram of the frustrated triangular lattice Heisenberg model coupled to dynamical phonons with energy $\omega$, for fixed spin-lattice coupling $\lambda=0.3$. Below $\omega_c$, we measure a finite-temperature phase transition $\widetilde{T}_c(\omega)$ from the U(1) Dirac spin liquid to spin-Peierls order (i.e., simultaneous valence bond solid order and lattice distortion). We use a variational method, where the spin wave function in the liquid regime is the fermionic state with alternating $0$ and $\pi$ fluxes (see upper-right inset for our gauge choice: on dotted bonds, the fermion hoppings have a negative sign) lu2016. The strengths of the $\langle S^z_i S^z_j\rangle$ correlations of nearest-neighbors are shown in the insets through the thickness/shading of the bonds. Finite-temperature responses are inferred from the finite-size behaviour of the system. We find that a conformal scaling Ansatz well describes the scaling of critical temperature, using which we measure the scaling dimension of monopoles $\Delta_\Phi=1.23(2)$.
• Figure 2: Static limit of lattice distortions and weak coupling physics. (a) Dimer-dimer correlations of the triangular-lattice DSL, fitted with a power-law $r^{-2\Delta_\Phi}$, $\Delta_\Phi=1.20(3)$ (b) Spin-Peierls temperature $\widetilde{T}_{\mathrm{SP}}$ as a function of the dimensionless spin-lattice coupling $\lambda$, for the 1D antiferromagnetic Heisenberg chain and the DSL ansatz for the triangular-lattice $J_1$-$J_2$ model (2D). See Supplementary Material for details of the fitted curves, obtained from the data collapse of Fig. \ref{['fig:collapses']}. (c,d) The energy gain per site $\delta\varepsilon$, as a function of static distortions, for various linear sizes $L$ in 1D and 2D (where $N=L\times L$ sites).
• Figure 3: Data collapse of the energy gain per site $\delta \varepsilon$ under static lattice distortion $g\, \delta u$. Results for (a) the Heisenberg chain and (b) the triangular lattice $J_1$--$J_2$ model are shown. We plot $L^d\delta\varepsilon / J_1$ against $g\, \delta u \, L^{d-\Delta}$ for dimension $d=D+1$ and optimise the scaling dimensions to get the best collapse, shown inset. Dotted and dashed line represent asymptotic fitting functions for the regimes of weak and strong coupling, i.e. at small and large values of $g\, \delta u$.
• Figure 4: (a) Measured critical phonon coupling strength $g_c$ of the U(1) DSL as a function of phonon frequency $\omega$ (dynamical phonons) and linear system size $L$. We produce a data collapse in (b) by plotting $g_c^2/(J_1 m) L^{5-2\Delta_\Phi}$ against $L\, \omega/J_1$.
• Figure 5: Panel (a): Brillouin zone of the triangular lattice with reciprocal lattice vectors and high symmetry points. Panel (b): lattice distortion giving the strongest response on the triangular lattice. Lattice sites are displaced according to Eq. \ref{['eq:dist_2d']} (with $\delta u >0$). The resulting 12-site supercell is delimited by the solid black lines. Different colors represent symmetry-inequivalent first-neighbor bonds, grouped into distinct classes according to their bond length after distortion. For each class of bonds, the value of the exchange interaction in presence of a finite static distortion $\delta u$ is reported. The variational parametrization of $\mathcal{H}_0$ follows the same symmetry of the distortions: hoppings with different absolute values are taken on the different classes of inequivalent bonds. The signs of the hoppings, instead, are kept equal to the ones of the DSL and are represented here by solid/dashed lines, which indicate positive/negative hoppings, respectively. Panel (c): energy gain per site $\delta \varepsilon$ as a function of the lattice distortions of Eq. \ref{['eq:dist_2d']}, for different signs of $\delta u$. Results for the $L=12$ lattice. The respective sites displacements are shown in the insets. The case $\delta u>0$ provides a larger energy gain beyond the weak-coupling regime of small distortions.