Fractional spin excitations and conductance in the spiral-staircase Heisenberg ladder

We investigate theoretically the spiral-staircase Heisenberg spin-$1/2$ ladder in the presence of antiferromagnetic long-range spin interactions and a uniform magnetic field. As a special case we also consider the Kondo necklace model. If the magnetizations of the two chains forming the ladder satisfy a certain resonance condition, involving interchain couplings as perturbations, the system is in a partially gapped magnetic phase hosting excitations characterized by fractional spins, whose values can be changed by the magnetic field. We show that these fractional spin excitations can be probed via the magnetization and by spin currents in a transport setup with a spin conductance that reveals the fractionalized spin. In some special cases, the spin conductance reaches universal values. We obtain our results analytically via bosonization techniques as well as numerically via density matrix renormalization group methods and find remarkable agreement between the two approaches.

Figure 1

(a) The spiral-staircase Heisenberg ladder is composed of two weakly coupled spin-$1/2$ (blue dots) chains aligned along the $x$ direction. The upper ($\tau =A$) and lower ($\tau =B$) chains are characterized by long-range exchange interactions within the chain, $J_r^{\tau }$, where $r$ is the range of interaction. The exchange interactions are assumed to be positive (antiferromagnetic), isotropic, and different in the two chains. Here, we indicate only the nearest-neighbor (yellow arrow) and the next-nearest-neighbor (orange arrow) interactions. The two chains are weakly coupled by an interchain interaction $J_{\perp }$ (brown arrow) and subjected to a uniform magnetic field $b$ (blue arrow). (b) Kondo necklace model with long-range spin interactions. Here, the spins belonging to chain $B$ are alternatingly displaced on the two sites of chain $A$. As a result, chain $B$ is split into two subchains with a doubled lattice constant, whose nearest-neighbor interaction $J_1^B$ is equivalent to the next-nearest-neighbor interaction $J_2^A$ of chain $A$.

Figure 2

(a) Luttinger liquid parameter $K_{\tau }$ computed by iDMRG simulations as function of magnetization $M_{\tau }$ for different values of the n.n.n. interaction $J_r^{\tau }$. The presence of long-range interaction in a single chain is crucial in order to achieve desirable values of $K_{\tau }$ lower than 0.5 (black dashed line). (b) Phase diagram for processes characterized by $(s_{\tau },s_{\overline{\tau }}=1)$ as functions of $K_{\tau }$ and $K_{\overline{\tau }}$ for $s_{\tau }>0$. For $K_{\tau }>0.5$, no processes induce fractional excitations ($s_{\tau }=0$) except for $|M_{\tau }|=|M_{\overline{\tau }}|$ (yellow). For smaller values of $K_{\tau }$, fractional phases with $s_{\tau }=2$ (green) or $s_{\tau }=3$ (blue) are stabilized. (c) For magnetizations $M_A$ and $M_B$, which follow Eq. (6) for the process ($s_A=2$, $s_B=1$) (blue line), we confirm numerically via DMRG (squares) fractional excitations $\mathrm{\Delta }\mathcal{P}=1/3$ for strong interchain interaction. (d) $\mathrm{\Delta }\mathcal{P}$ is computed as a function of interchain coupling by DMRG at different values of magnetizations taken from panel (c): the colors of the lines correspond to the colors of the squares in panel (c). The interaction parameters are $J_2^A=0.5J_1^A$, $J_2^B=0.1J_1^A$, and $J_2^B=0.1J_1^A$.

Figure 3

Correlation functions for the chain $\tau$ and corresponding fitted Luttinger liquid parameter $K_{\tau }$ for a fixed magnetization $M_{\tau }$ and next-nearest-neighbor interaction strength $J_2^{\tau }$. (a) In-plane correlation function ${\mathcal{C}}_{xy}$. The parameters are $J_2^{\tau }=0.2J_1^{\tau }$ and $M^{\tau }=0.25$. (b) Out-of-plane correlation function ${\mathcal{C}}_z$. The parameters are $J_2^{\tau }=0.2J_1^{\tau }$ and $M^{\tau }=0.25$. (c) In-plane correlation function ${\mathcal{C}}_{xy}$. The parameters are $J_2^{\tau }=0.35J_1^{\tau }$ and $M^{\tau }=0.125$. (d) Out-of-plane correlation function ${\mathcal{C}}_z$. The parameters are $J_2^{\tau }=0.35J_1^{\tau }$ and $M^{\tau }=0.125$. The choice of next-nearest-neighbor interaction strength $J_2^{\tau }$ corresponds to the one made for Fig. 2 in the main text.

Figure 4

The phase diagram for the process ${\mathcal{H}}_{s_{\tau }{s}_{\overline{\tau }}}$ as a function of LL parameters of chains $\tau$ and $\overline{\tau }$ for different values of $s_{\tau }$ and $s_{\overline{\tau }}$.

Figure 5

Color plot of the dynamical spin structure factor $\mathcal{S}(q,\omega )$ for positive energies $\omega$ and wave vector $q$ for the illustrative example of $s_A=2$, $s_B=1$ (giving fractional spin $1/3$). The function is symmetric in $q$ and $\omega$. The values of $q_0$ and ${\omega }_0$ are defined in the text. The dark blue background corresponds to values of $\mathcal{S}(q,\omega )$ much smaller than those of the green and light blue curves, given by the poles ${\omega }_{\pm }$ [Eq. (B14)]. Here, $\delta =2{\omega }_0$ is the gap, which depends on $s_{\tau }$; see Eq. (B16). The other parameters are chosen as $K_A=0.3$, $K_B=0.36$, $\gamma =1.5{\omega }_0$, and $u_A/u_B=3.15$.

Figure 6

Setup to measure the fractional spin conductance. In the left and right leads (blue regions) different magnetic fields ($b^1=b^L+\mathrm{\Delta }b/2$ and $b^2=b^L-\mathrm{\Delta }b/2$) are applied.

Figure 7

Setup to measure the fractional spin conductance. In the left and right leads (blue regions) different uniform magnetic fields with Zeeman energies $b^1=b^L+\mathrm{\Delta }b/2$ and $b^2=b^L-\mathrm{\Delta }b/2$ are applied. The leads are uncoupled isotropic spin-$1/2$ chains. If one assumes $b^L\sim 0$, then the LL parameters in the chains are $K_A^L=K_B^L\equiv K^L=\frac{1}{2}$, regardless of the strength of long-range interactions.