Intertwined spin and charge dynamics in one-dimensional supersymmetric t-J model

Abstract

Following the Bethe ansatz we determine the dynamical spectra of the one-dimensional supersymmetric t-J model. A series of fractionalized excitations are identified through two sets of Bethe numbers. Typical patterns in each set are found to yield wavefunctions containing elementary spin and charge carriers, manifested as distinct boundaries of the collective excitations in the spectra of single electron Green functions. In spin channels, gapless excitations fractionalized into two spin and a pair of postive and negative charge carriers, extending to finite energy as multiple continua. These patterns connect to the half-filling limit where only fractionalized spinons survive. In particle density channel, apart from spin-charge fractionalization, excitations involving only charge fluctuations are observed. Furthermore, nontrivial Bethe strings encoding bound state structure appear in channels of reducing or conserving magnetization, where spin and charge constituents can also be identified. These string states contribute significantly even to the low-energy sector in the limit of vanishing magnetization.

Figures (5)

• Figure 1: ($a$) Examples of BN patterns. The ground state configurations for both $\{I_a^n\}$ and $\{J\}$ possess minimal total absolute value. $n \psi$ are created by choosing desired number of BNs out of ground state occupations combined with $n$ innermost unoccupied BNs, either from the left, right or both sides. $n \psi^*$ are created by replacing $n$ outermost occupied BNs by unoccupied BNs. $n\psi\psi^*$ corresponds to moving $n$ occupied BNs to unoccupied positions. These classifications can overlap with each other, e.g., $1\psi_c$ is a subset of $1\psi_c\psi_c^*$. ($b$,$c$) Examples of the energy and momentum of Bethe states ($b$) solved from BN patterns in ($c$). Each column in ($c$) stands for a set of BNs for one state (here $L=30$, $N_\downarrow=8$, $N_h=6$). The colored dots and stars in ($c$) correspond to the states of the same color in ($b$), with the columns in ($c$) ordered according to the clockwise sequence of the points in ($b$) starting from the gapless point. Part of the $c^*$ and $s$ bands are also illustrated.
• Figure 2: Dynamic structure factors at $g=0$ for different fillings in each column. $\alpha=x,+,-$ and $\beta=\uparrow,\downarrow$ in the second through fourth columns. In the single hole case shown in the first column, $\alpha=z$ and $\beta=\uparrow$, with other channels exhibiting similarly as the SU$(2)$ symmetry is slightly broken. $k_F^\uparrow$ and $k_F^\downarrow$ differ by $\pi/L$ in this case, and we omit the difference in the denotations. Some featured single particle dispersions of $\mathscr{L}_1$ type are denoted by black and gray text and arrows, representing branches that reach or are gapped from the lowest energy in $\mathscr{L}_1$ region. The blue and light blue ones illustrate similarly for $\mathscr{L}_2$ case. Markers on the horizontal axis illustrate the gapless points.
• Figure 3: DSFs for $L=60$, $n_e=0.9$ ground states with different magnetization in each row. We define $\overline{k}\equiv\pi-k$. The same notations are applied as in Fig. \ref{['fig:M0']}, and purple color is used in $\mathscr{L}_3$ case.
• Figure 4: DSFs for ground state with one hole and $L=60$. Ground state with the same magnetization is illustrated by the same color. The same notations are applied as in Figs. (\ref{['fig:M0']},\ref{['fig:N0p9']}). Closed gapless points separated by $k=2\pi/L$ is illustrated as one point in the notations for brevity.
• Figure S1: Example of DSFs contributed from $\mathscr{L}_n$ states with size $L=60$. Percentage in each figure shows the zero sum saturation of spectral weight contributed by $\mathscr{L}_n$ in the total spectrum.