Spin-phonon interactions on the kagome lattice: Dirac spin liquid versus valence-bond solids

We investigate the impact of spin-phonon coupling on the $S=1/2$ Heisenberg model on the kagome lattice. For the pure spin model, there is increasing evidence that the low-energy properties can be correctly described by a Dirac spin liquid, in which spinons with a conical dispersion are coupled to emergent gauge fields. Within this scenario, the ground-state wave function is well approximated by a Gutzwiller-projected fermionic state [Ran, Hermele, Lee, and Wen, Phys. Rev. Lett. 98, 117205 (2007)]. However, the existence of $U\left(1\right)$ gauge fields may naturally lead to instabilities when small perturbations are included. Phonons are ubiquitous in real materials and may play a relevant role in the determination of the actual physical properties of the kagome antiferromagnet. Therefore, we perform a step forward in this direction, including phonon degrees of freedom (at the quantum level) and applying a variational approach based upon Gutzwiller-projected fermionic Ansätze. Our results suggest that the Dirac spin liquid is stable for small spin-phonon couplings, while valence-bond solids are obtained at large couplings. Even though different distortions can be induced by the spin-phonon interaction, the general aspect is that the energy is lowered by maximizing the density of perfect hexagons in the dimerization pattern.

Figure 1

Lattice vectors employed to construct different supercells for the variational Ansätze. Taking the nearest-neighbor distance as unit, the red vectors are defined as ${\vec{a}}_1=(2,0)$ and ${\vec{a}}_2=(1,\sqrt{3})$ (primitive vectors). The blue vectors are ${\vec{b}}_1=2{\vec{a}}_1-{\vec{a}}_2$ and ${\vec{b}}_2=2{\vec{a}}_2-{\vec{a}}_1$. The green vectors are ${\vec{c}}_1={\vec{a}}_1+{\vec{a}}_2$ and ${\vec{c}}_2={\vec{a}}_2-{\vec{a}}_1$.

Figure 2

Results of the calculations on the ${(6,6)}_a$ cluster ($N=108$ sites). In panel (a) the energies (per site) of the Dirac spin liquid (SL) and different valence-bond ordered states are shown as a function of the spin-phonon coupling $g$. The zero-point energy of the phonon Hamiltonian, $E_0^{\mathrm{ph}}=N\omega$, has been subtracted. The patterns of distortions associated with the various valence-bond ordered states are shown in panels (b), (c), and (d), as obtained within the ${(2,1)}_a, {(3,3)}_a$, and ${(2,3)}_a$ supercells, respectively. The color of the bonds between sites $i$ and $j$ represents the value of the spin-spin correlations $\langle S_i^z{S}_j^z\rangle$ (at $g=0.3$). The dashed lines delimit the supercells.

Figure 3

The same as in Fig. 2 for calculations on the ${(12,12)}_a$ cluster ($N=432$ sites). Panel (a) shows the energies (per site) of the Dirac spin liquid (SL) and different valence-bond solids with perfect hexagons. Panels (b) and (c) show the distortions and spin-spin correlations (at $g=0.3$) of the ${(2,2)}_b$ and ${(3,2)}_c$ supercells, respectively.

Figure 4

Fermionic hoppings in the minimal variational Ansatz for the ${(2,2)}_b$ valence-bond solid. Bonds of different colors correspond to (four) independent absolute values of the hopping parameters. Solid and dashed lines denote positive and negative signs of the hopping, respectively. The fluxes threading the hexagonal plaquettes are reported and compatible with those observed in Ref. [23]. Perfect hexagons can be recognized as the ones with zero flux.

Figure 5

Energy loss due to forcing a dimerization of the perfect hexagons in the variational Ansatz of the ${(2,2)}_b$ valence-bond solid (Fig. 4). The total energy loss (red squares) is plotted alongside the contribution due to the free phonon Hamiltonian (blue circles), for two inequivalent dimerization patterns (illustrated in the insets). The results are obtained on the ${(4,4)}_b$ cluster ($N=144$ sites) for $g=0.3$.

Figure 6

Energy landscape of the pinwheel valence-bond solid Ansatz [29] as a function of the tuning parameter $\delta t$. The three independent hoppings defining the variational state are shown in the upper inset. Different colors refer to different absolute values, with red/black bonds indicating $|t_{i,j}|=1\pm \delta t$. Solid and dashed lines denote, respectively, positive and negative hoppings, which reproduce the characteristic fluxes of the Dirac spin liquid. The lower inset shows the lattice distortions and spin-spin correlations (analogously to Figs. 2 and 3) in the minimum of the variational energy, for $g=0.3$. The results are obtained on the ${(6,6)}_a$ cluster ($N=108$ sites).