Localized and propagating excitations in gapped phases of spin systems with bond disorder

Using the conventional $T$-matrix approach, we discuss gapped phases in one-, two-, and three-dimensional (3D) spin systems (both with and without a long-range magnetic order) with bond disorder and with weakly interacting bosonic elementary excitations. This work is motivated by recent experimental and theoretical activity in spin-liquid-like systems with disorder and in the disordered interacting boson problem. In particular, we apply our theory to both paramagnetic low-field and fully polarized high-field phases in dimerized spin-$\frac{1}{2}$ systems and in integer-spin magnets with large single-ion easy-plane anisotropy $\mathcal{D}$ with disorder in exchange coupling constants (and/or $\mathcal{D}$). The elementary excitation spectrum and the density of states are calculated in the first order in defects concentration $c\ll 1$. In 2D and 3D systems, the scattering on defects leads to a finite damping of all propagating excitations in the band except for states lying near its edges. We demonstrate that the analytical approach is inapplicable for states near the band edges and our numerical calculations reveal their localized nature. We find that the damping of propagating excitations can be much more pronounced in considered systems than in magnetically ordered gapless magnets with impurities. In 1D systems, the disorder leads to localization of all states in the band, while those lying far from the band edges (short-wavelength excitations) can look like conventional wave packets.

Figure 1

Phase diagrams of (quasi-)3D dimerized spin-$\frac{1}{2}$ systems and spin-1 magnets with large single-ion easy-plane anisotropy $\mathcal{D}$ in magnetic field $H$. (a) Systems without defects with a canted magnetic ordering inside the dome. (b) Systems with a small fraction of bonds with strengthen and/or weaken exchange coupling constants (and/or $\mathcal{D}$ value). Bose-glass phases are denoted as BG.

Figure 2

1D systems with imperfect bonds shown by dashed lines. (a) Spin-$\frac{1}{2}$ ladder with dimers on rungs (shown in bold) with modified intradimer exchange constant $\mathcal{J}$ at rung 1 and modified values of exchange coupling constants between spins from dimer 1 and neighboring dimers 0 and 2. (b) Integer spin chain with modified value of the single-ion easy-plain anisotropy $\mathcal{D}$ at site 1 and modified value of exchange coupling constant at bonds 0-1 and 1-2.

Figure 3

Correction to quasiparticles energy and their damping in 1D systems given by Eqs. (32) and found numerically for disorder in $\mathcal{J}$ or $\mathcal{D}$ only (the extrapolation is carried out of numerical data for finite systems containing $L$ unit cells to thermodynamical limit as explained in Sec. 3b). Shaded regions mark areas in which the imaginary parts of one-particle Green's functions ${\chi }^{\prime \prime }(k,\omega )$ found numerically using Eq. (28) do not have a Lorentzian shape and in which our analytical results are invalid [i.e., inequalities (34) do not hold]. Insets in planes (b) and (d) show ${\chi }^{\prime \prime }(k,\omega )$ for some fixed momenta, where solid lines represent results of data fitting by Lorentzians. Most pronounced anomalies in numerical data for “stronger” impurities [planes (c) and (d)] are interpreted as a result of coherent scattering by defects of quasiparticles with momenta denoted by vertical lines (see the text).

Figure 4

DOS of 1D systems with disorder in $\mathcal{J}$ or $\mathcal{D}$ only, where $\delta E=E-\Delta -a\left|J\right|$. DOS in the $T$-matrix approach is given by Eq. (25) (it is almost indistinguishable on the plots from the pure system DOS). Most pronounced anomalies in numerical data found for $L=3000$ are interpreted as a result of coherent scattering by defects of quasiparticles with energies denoted by vertical lines (see the text).

Figure 5

The inverse participation ration (IPR) given by Eq. (29) averaged over disorder realizations as it is described in Sec. 3b. $\sigma \left(\mathrm{IPR}\right)$ is the mean-square deviation of IPR from its mean value. (a), (b) and (c), (d) slides are for particular states inside the band far from its edges in 1D and 2D systems, respectively. The states' energies $\delta E$ are measured from the band center [see Figs. 4 and 8, correspondingly]. Insets show histograms of IPR distributions in disorder realizations.

Figure 6

Spin-1/2 dimerized bilayer with imperfect bonds. Notations are the same as in Fig. 2.

Figure 7

(a) Spatial distribution of defects with $c=0.1$, $u=3a\left|J\right|$, and $u_1=0$ in 2D system with the size $120\times 120$ unit cells. (b)–(j) Numerically found color plots of wave functions amplitudes for the Hamiltonian of this disordered system which correspond to indicated eigenvalues $E$. Panels (b)–(e) give a picture of energy levels near the band bottom, panels (f)–(h) illustrate the band top, and panels (i) and (j) describe the impurity band corresponding to the localized level in the first order in $c$ (see also Fig. 8 for DOS found for the same parameters). All states in the impurity band are localized. $\Delta$ is the gap value for the particular disorder realization.

Figure 8

DOS of 2D systems with disorder in $\mathcal{J}$ or $\mathcal{D}$ only, where $\delta E=E-\Delta -2a\left|J\right|$, $c=0.1$, $u=3a\left|J\right|$, and $u_1=0$ (cf. Fig. 4). Numerical results are obtained for the system size $100\times 100$ unit cells.

Figure 9

3D spin-$\frac{1}{2}$ dimer system with imperfect bonds. Notations are the same as in Figs. 2 and 6.

Figure 10

DOS of 3D systems, where $\delta E=E-\Delta -3a\left|J\right|$, $c=0.1$, $u=3a\left|J\right|$, and $u_1=0$. Solid and dashed lines are for pure and discorded systems, respectively.