Massive spinons in $S=1/2$ spin chains: Spinon-pair operator representation

Spinons are among the generic excitations in one-dimensional spin systems; they can be massless or massive. The quantitative description of massive spinons poses a considerable challenge in spite of various variational approaches. We show that a representation in terms of hopping and Bogoliubov spinon processes, which we call “spinon-pair” operators, and their combination is possible. We refer to such a representation as second quantized form. Neglecting terms which change the number of spinons yields the variational results. Treating the bilinear and quartic terms by continuous unitary transformations leads to considerably improved results. Thus, we provide the proof of principle that systems displaying massive spinons as elementary excitations can be treated in second quantization based on spinon-pair representation.

Figure 1

The spinon states with (a) $S_{\mathrm{tot}}=0$ and (b) $S_{\mathrm{tot}}=1$ on a 6-site cluster. Each triplet bond (dashed line) can have the flavor $p=x,y,z$. The $S_{\mathrm{tot}}=1$ states are constructed from $S_{\mathrm{tot}}=0$ states by replacing one singlet bond with a triplet bond; no nested triplet bond is allowed as explained in the main text.

Figure 2

Analysis of the commutator $[{\mathcal{S}}_{i,i+d}^{†},{\mathcal{H}}_{j,j+d^{\prime }}^{\phantom{†}}]$ in different cases. The double-headed arrow creates two spinons at the two ends $(i$ and $i+d)$ and denotes the singlet operator ${\mathcal{S}}_{i,i+d}^{†}$. The hopping operator ${\mathcal{H}}_{j,j+d^{\prime }}^{\phantom{†}}$ is shown by a dashed arrow indicating that a spinon hops from site $j$ to $j+d^{\prime }$.

Figure 3

The spin gap ${\mathrm{\Delta }}_s$ versus the frustration $\alpha$. The various CUT truncation schemes are compared to the variational results by Brehmer et al. [27] and to the DMRG results by Chitra et al. [63] and White and Affleck [64].

Figure 4

The momentum $(\frac{\pi }{2}-k^{*})$ versus the frustration $\alpha ;k^{*}$ is the momentum at which the minimum of the spinon dispersion is located.

Figure 5

The low-energy spectrum of the $J_1\text{−}{J}_2$ Heisenberg model (1) with (a) an odd and (b) an even number of sites at frustration $\alpha =0.8$. Panel (a) displays the 1-spinon dispersion and the 3-spinon continuum and panel (b) denotes the 2-spinon continuum. The continua are constructed by energy and momentum conservation from the spinon dispersion with $d_{\max }=30$. The results for $d_{\max }=25$ are the same within the linewidth.