Dynamic structure factor of Heisenberg bilayer dimer phases in the presence of quenched disorder and frustration

We investigate the influence of quenched disorder on the dynamic structure factor of Heisenberg bilayers on the square, triangular, and kagome lattice in the quantum paramagnetic phase. Perturbative continuous unitary transformations and white graphs are employed to calculate the one-triplon contribution up to high orders in perturbation about the dimer limit for bimodal and continuous disorder. For the square lattice we find that the lifetime of the gap mode is increased by stronger quantum correlations, while stronger disorder effects are observed for the triangular lattice due to geometric frustration. For intradimer disorder, in-band energy gaps are observed for both lattices which can be understood in terms of a level repulsion on dimers with low and high intradimer exchange that are close in energy at the momentum where the in-band gap opens. For the highly frustrated kagome lattice disorder even decreases the gap energy. In addition, the localization length of the low-energy flat band is increased up to order 7 in perturbation theory. The interplay of quenched disorder, geometric frustration, and strong correlations leads therefore to rich structures in the dynamical structure factor of two-dimensional quantum magnets.

Figure 1

Illustration of the Heisenberg bilayer on the (a) square, (b) triangular, and (c) kagome lattice. Blue-shaded boxes illustrate the unit cell in each case.

Figure 2

The DSF ${\mathcal{S}}_{-}(k,\omega )$ is shown for bimodal intradimer disorder for the square lattice with $J^{\parallel }=0.2$ and $\mathrm{\Delta }{J}^{\perp }=\pm 0.2$ for $p=0.1$ in (a), $p=0.5$ in (b), and $p=0.9$ in (c). The maximum intensity of 80 in (a), 35 in (b), and 250 in (c) was truncated at 45 in (a), at 20 in (b), and at 100 in (c) to make more details visible in the plots.

Figure 3

For the bimodal intradimer disorder in the square lattice and for the probabilities $p=0.3,0.5,0.7$ the overlap $n$ of the eigenfunctions with dimer states of value $J_1^{\perp }=0.8$ is shown as a function of $\omega$. Up to the transition energy region the overlap is larger than 0.8 for $p=0.3,0.5,0.7$. In contrast, for larger $\omega$ the overlap with $J_1^{\perp }=0.8$ dimer states is smaller than 0.2.

Figure 4

The DSF ${\mathcal{S}}_{-}(k,\omega )$ is shown for bimodal intradimer disorder in the triangular lattice with $J^{\parallel }=0.15$ and $\mathrm{\Delta }{J}^{\perp }=\pm 0.15$ for $p=0.1$ in (a), $p=0.5$ in (b), and $p=0.9$ in (c). The maximum intensity of 90 in (a) and 200 in (c) was truncated at 70 in (a) and at 150 in (c) to make more details visible in the plots.

Figure 5

The DSF ${\mathcal{S}}_{-}(k,\omega )$ is shown for bimodal intradimer disorder in the kagome lattice with $J^{\parallel }=0.2$ and $\mathrm{\Delta }{J}^{\perp }=\pm 0.2$ for $p=0.1$ in (a), $p=0.5$ in (b), and $p=0.9$ in (c). The maximum intensity of 250 in (a) and 250 in (c) was truncated at 90 in (a) and at 70 in (c) to make more details visible in the plots.

Figure 6

The DSF ${\mathcal{S}}_{-}(k,\omega )$ is shown for bimodal intradimer disorder in the square lattice with $J^{\parallel }=0.2$ and $\mathrm{\Delta }{J}^{\perp }=\pm 0.2$ for $p=0.5$ in blue and in red for a bimodal Gaussian distribution with agreement in the first four moments with the bimodal distribution. In (a) $\mathit{k}=(0,0)$, in (b) $\mathit{k}=(0,\pi )$, and in (c) $\mathit{k}=(\pi ,\pi )$.

Figure 7

The DSF ${\mathcal{S}}_{-}(k,\omega )$ is shown for bimodal interdimer disorder in the square lattice with $J^{\perp }=1$, $J_1^{\parallel }=0.1$, and $J_2^{\parallel }=0.3$ for $p=0.5$ in (a) and for $p=0.1,0.3,0.5,0.7,0.9$ and momenta $\mathit{k}=(0,0)$ in (b), $\mathit{k}=(0,\pi )$ in (c), and $\mathit{k}=(\pi ,\pi )$ in (d).

Figure 8

The DSF ${\mathcal{S}}_{-}(k,\omega )$ is shown for bimodal interdimer disorder in the triangular lattice with $J^{\perp }=1$, $J_1^{\parallel }=1/15$, and $J_2^{\parallel }=1/5$ for $p=0.5$ in (a) and for $p=0.1,0.3,0.5,0.7,0.9$ and momentas $\mathit{k}=(0,0)$ in (b), $\mathit{k}=(2\pi /3,2\pi /3)$ in (c), and $\mathit{k}=(2\pi /3,4\pi /3)$ in (d).

Figure 9

The DSF ${\mathcal{S}}_{-}(k,\omega )$ is shown for bimodal interdimer disorder in the kagome lattice with $J^{\perp }=1$, $J_1^{\parallel }=0.1$, and $J_2^{\parallel }=0.3$ for $p=0.1$ in (a), $p=0.5$ in (b), and $p=0.9$ in (c).

Figure 10

The DSF ${\mathcal{S}}_{-}(k,\omega )$ is shown for bimodal interdimer disorder in the square lattice with $J^{\perp }=1$, $J_1^{\parallel }=0.1$, and $J_1^{\parallel }=0.3$ for $p=0.5$ in blue. In red it is shown for a Gaussian distribution and in yellow for a box distribution, with agreement in the first two moments with the bimodal distribution. In (a) $\mathit{k}=(0,0)$, in (b) $\mathit{k}=(0,\pi )$, and in (c) $\mathit{k}=(\pi ,\pi )$.

Figure 11

The DOS is shown for bimodal intradimer disorder, $p=0.1,0.3,0.5,0.7,0.9$ and the corresponding disorder configurations of the main body of the paper in (a) for the square, in (b) for the triangular, and in (c) for the kagome lattice.

Figure 12

The DOS is shown for bimodal interdimer disorder, $p=0.1,0.3,0.5,0.7,0.9$, and the corresponding disorder configurations of the main body of the paper in (a) for the square, in (b) for the triangular, and in (c) for the kagome lattice.

Figure 13

The IPR is shown for bimodal intradimer disorder, $p=0.1,0.3,0.5,0.7,0.9$, and the corresponding disorder configurations of the main body of the paper in (a) for the square, in (b) for the triangular, and in (c) for the kagome lattice.

Figure 14

The IPR is shown for bimodal interdimer disorder, $p=0.1,0.3,0.5,0.7,0.9$, and the corresponding disorder configurations of the main body of the paper in (a) for the square, in (b) for the triangular, and in (c) for the kagome lattice.

Figure 15

The DSF ${\mathcal{S}}_{-}(k,\omega )$ and its finite-size scaling behavior is shown for bimodal intradimer disorder, $p=0.1,0.3,0.5,0.7,0.9$, and the corresponding disorder configurations of the main body of the paper in (a) for the square, in (b) for the triangular lattice, and in (c) for the kagome lattice. The momentum $\mathit{k}$ is the gap momentum in the corresponding lattices, i.e., $\mathit{k}=(\pi ,\pi )$ for the square, $\mathit{k}=(2\pi /3,4\pi /3)$ for the triangular, and $\mathit{k}=(2\pi /3,4\pi /3)$ for the kagome lattice.

Figure 16

The DSF ${\mathcal{S}}_{-}(k,\omega )$ and its finite-size scaling behavior is shown for bimodal interdimer disorder, $p=0.1,0.3,0.5,0.7,0.9$, and the corresponding disorder configurations of the main body of the paper in (a) for the square, in (b) for the triangular lattice, and in (c) for the kagome lattice. The momentum $\mathit{k}$ is the gap momentum in the corresponding lattices, i.e., $\mathit{k}=(\pi ,\pi )$ for the square, $\mathit{k}=(2\pi /3,4\pi /3)$ for the triangular, and $\mathit{k}=(2\pi /3,4\pi /3)$ for the kagome lattice.

Figure 17

The DSF ${\mathcal{S}}_{+}(k,\omega )$ and its finite-size scaling behavior is shown for bimodal interdimer disorder, $p=0.1,0.3,0.5,0.7,0.9$, and the corresponding disorder configurations of the main body of the paper in (a) for the square, in (b) for the triangular lattice, and in (c) for the kagome lattice. The momentum $\mathit{k}$ is the gap momentum in the corresponding lattices, i.e., $\mathit{k}=(\pi ,\pi )$ for the square, $\mathit{k}=(2\pi /3,4\pi /3)$ for the triangular, and $\mathit{k}=(2\pi /3,4\pi /3)$ for the kagome lattice.

Figure 18

(a), (b) The DSF ${\mathcal{S}}_{-}(k,\omega )$ is shown for bimodal intradimer disorder in the square lattice. The disorder is identical to that in the main body of the paper. In (a) $p=0.3$ and in (b) $p=0.7$. (c), (d) The DSF ${\mathcal{S}}_{-}(k,\omega )$ is shown for bimodal intradimer disorder in the triangular lattice. The disorder is identical to that in the main body of the paper. In (c) $p=0.3$ and in (d) $p=0.7$.

Figure 19

(a), (b) The DSF ${\mathcal{S}}_{-}(k,\omega )$ is shown for bimodal intradimer disorder in the kagome lattice. The disorder is identical to that in the main body of the paper. In (a) $p=0.3$ and in (b) $p=0.7$. (c), (d) The DSF ${\mathcal{S}}_{-}(k,\omega )$ is shown for bimodal interdimer disorder in the kagome lattice. The disorder is identical to that in the main body of the paper. In (c) $p=0.3$ and in (d) $p=0.7$.

Figure 20

The DSF ${\mathcal{S}}_{+}(k,\omega )$ is shown for bimodal interdimer disorder, $p=0.5$, and the corresponding disorder configurations of the main body of the paper in (a) for the square, in (b) for the triangular, and in (c) for the kagome lattice.

Figure 21

The DSF ${\mathcal{S}}_{+}(k,\omega )$ is shown for bimodal interdimer disorder, $p=0.1,0.3,0.5,0.7,0.9$, and the corresponding disorder configurations of the main body of the paper in (a) for the square, in (b) for the triangular lattice, and in (c) for the kagome lattice. The momentum $\mathit{k}$ is the gap momentum in the corresponding lattices, i.e., $\mathit{k}=(\pi ,\pi )$ for the square, $\mathit{k}=(2\pi /3,4\pi /3)$ for the triangular, and $\mathit{k}=(2\pi /3,4\pi /3)$ for the kagome lattice.