Breathers and Raman scattering in a two-leg ladder with staggered Dzyaloshinskii-Moriya interaction

Recent experiments have revealed the role of the staggered Dzyaloshinskii-Moriya interaction in the magnetized phase of an antiferromagnetic spin-$1∕2$ two-leg ladder compound under a uniform magnetic field. We derive a low-energy effective field theory describing a magnetized two-leg ladder with a weak staggered Dzyaloshinskii-Moriya interaction. This theory predicts the persistence of the spin gap in the magnetized phase, in contrast to standard two-leg ladders, and the presence of bound states in the excitation spectrum. Such bound states are observable in Raman scattering measurements. These results are then extended to intermediate Dzyaloshinskii-Moriya interaction using exact diagonalizations.

Figure 1

Sketch of the Raman intensity for polarization parallel to the legs $\left(XX\right)$. The arrows represent the position of the breather $\delta$ peaks. The intensity in the peak, proportional to its height, decreases with increasing breather index. The continuum starts at a frequency equal to twice the mass of the lightest breather. Above the threshold, resonances are expected instead of $\delta$ functions (beyond a SG analysis).

Figure 2

Sketch of the Raman intensity for polarization parallel to the rungs $\left(YY\right)$. The arrows represent the positions of the breather $\delta$ peaks. The intensity in the peak, proportional to its height, decreases with increasing breather index. In contrast to the case of polarization along the legs, only odd breathers contribute to the Raman intensity. The continuum starts at a frequency equal to the sum of the masses of the lightest odd and even breathers, which is higher than in the case of polarization along the legs.

Figure 3

(Color online) Magnetization (a) and gap (b) vs (reduced) magnetic field for finite $D=0.1$ and 0.15, compared to the $D=0$ case, computed on a $2\times 12$ ladder. Note that the $h=h_{c,1}$ singularity at $D=0$ disappears for finite $D$. (The behavior of the gap at $D=0$ is schematic.) All energies are in units of $J_{\perp }$.

Figure 4

(Color online) Low-energy spectra in the four different $(p,i)$ symmetry sectors for $2\times L$ ladders, $L$ ranging from $L=4$ to 14, according to color (symbol) codes shown on the plots, and for $D=0.1$ (a) and 0.15 (b). All energies are measured from the GS energy $(E=0)$. Full (LAPACK) diagonalization (small size $L=4,6$), Lanczos, and/or Davidson ED $(L=8,10,12,14)$ have been performed (Ref. 87). The tentative onsets of the two-particle continua are indicated by arrows.

Figure 5

(Color online) Ratio of the two quasidegenerate excited states in the (+, +) and (−, +) sectors (labeled as $S$ in Fig. 4) as a function of inverse system size for different values of $D$ (as indicated on the plot).

Figure 6

(Color online) $R_{1,2}$ (a) and $R_{1,3}$ (b) of Eq. (38) as functions of $D$ for different system sizes (as indicated on the plot). The expected $L\to \infty$ extrapolation is tentatively sketched as a dashed line. The magnetic field is such that $m=0.5m_{\mathrm{sat}}$ and $D$ is measured in units of $J_{\perp }$.

Figure 7

Raman spectra, computed on an $L=14$ cluster, for the three different polarizations $XX$, $YY$, and $X^{\prime }{Y}^{\prime }$ (see text) and for $D=0.15$ and a magnetic field such that $m=0.5m_{\mathrm{sat}}$. The spurious $\omega =0$ peak has been subtracted (see text). Note that a small imaginary part $ϵ=0.01$ in the frequency has been used to give a small width to the low-energy $\delta$ peaks. The corresponding weights of these peaks are shown in Fig. 8 as a function of system size. Insets: Relative weight of the $S$ state with respect to the $B_1$ one vs $D$, compatible with a vanishing $S$ intensity as $D\to 0$.

Figure 8

(Color online) Finite-size scaling of the Raman spectral weights of the soliton and the breathers at $D=0.15$ and for a magnetic field such that $m=0.5m_{\mathrm{sat}}$. All light polarizations $XX$, $YY$, and $X^{\prime }{Y}^{\prime }$ are shown. A logarithmic scale is used to reveal the tiny weights in the $YY$ polarization. For $XX$ and $X^{\prime }{Y}^{\prime }$ polarizations, a sizable fraction of the overall weight (normalized to 1) is located at higher energies (not shown).

Figure 9

(Color online) Evolution of the gap $\left(M_1\right)$ as a function of $D$ for a magnetic field such that $m=0.5m_{\mathrm{sat}}$ (logarithmic scale). The results for several lengths are reported and the extrapolation for infinite size is consistent with a $D^{3∕5}$ behavior (continuous line). The bosonization results, shown as a dashed line, differ only by a multiplicative factor of order 1.25. All energies are in units of $J_{\perp }$.