Disordered quantum spin chains with long-range antiferromagnetic interactions

We investigate the magnetic susceptibility $\chi \left(T\right)$ of quantum spin chains of $N=1280$ spins with power-law long-range antiferromagnetic couplings as a function of their spatial decay exponent $\alpha$ and cutoff length $\xi$. The calculations are based on the strong disorder renormalization method which is used to obtain the temperature dependence of $\chi \left(T\right)$ and distribution functions of couplings at each renormalization step. For the case with only algebraic decay $\left(\xi =\infty \right)$ we find a crossover at ${\alpha }^{*}=1.066$ between a phase with a divergent low-temperature susceptibility $\chi (T\to 0)$ for $\alpha >{\alpha }^{*}$ to a phase with a vanishing $\chi (T\to 0)$ for $\alpha <{\alpha }^{*}$. For finite cutoff lengths $\xi$, this crossover occurs at a smaller ${\alpha }^{*}\left(\xi \right)$. Additionally, we study the localization of spin excitations for $\xi =\infty$ by evaluating the distribution function of excitation energies, and we find a delocalization transition that coincides with the opening of the pseudogap at ${\alpha }_c={\alpha }^{*}$.

Figure 1

Left: Sketch of electronic orbitals at low doping concentration $n_{\mathrm{D}}$. All states are localized and magnetic, as indicated by red arrows. Right: At larger $n_{\mathrm{D}}$, states at Fermi energy are delocalized, coexisting with localized magnetic states.

Figure 2

Magnetic susceptibility, normalized by ${\chi }_0\equiv \chi (T=J_0)$, for an $XX$ chain with $N=1280$ randomly placed spins, interacting via antiferromagnetic long-range couplings, Eq. (2), chain length $L/a=100N$, and $\alpha =0.6,\cdots ,2.0$. The cutoff length is set to $\xi =\infty$. Continuous lines: Fits to $\chi \sim T^{1/z-1}$ with finite $z$ at low temperatures. Inset: Susceptibility for $\alpha =10.0$ along with the power law with $z=9.36$ (green line) and the IRFP result (red line).

Figure 3

Dynamical exponent $z$ extracted by fitting the low-temperature susceptibility in Fig. 2 to Eq. (7) as a function of the power $\alpha$ (squares). These numerical results are then fit to Eq. (8) (dashed line), which allows us to extract the crossover value ${\alpha }^{*}=1.066\pm 0.002$.

Figure 4

Width $W_1$ of the distribution function of nearest-neighbor couplings as a function of the fraction of remaining spins. The negative logarithmic in base two is used to have an equally spaced horizontal variable that grows as the number of spins decreases. All values have been normalized by the width $W_1^0\left(\alpha \right)$ of the initial distribution and the parameters are kept as in Fig. 2. Inset: $W_1/W_1^0$ for a simple nearest-neighbor model with uniformly distributed couplings (crosses) known to flow to the IRFP. The numerical results for $\alpha =20.0$ (black circles) are included for comparison purposes.

Figure 5

Distribution of the lowest excitation gap ${\epsilon }_1$ scaled by its mean value, $s={\epsilon }_1/\langle {\epsilon }_1\rangle$, for $\xi =\infty$ and $\alpha =0.6,0.8,\cdots ,2.0$. The remaining parameters are as in Fig. 2.

Figure 6

Log-log plot of the magnetic susceptibility for $\alpha =d=1$ and $\xi /L=1,1/2,\cdots ,1/64$. The remaining parameters are kept as in Fig. 2, and the susceptibility is rescaled in the same manner.

Figure 7

Logarithmic plot of the susceptibility for a fixed and finite cutoff length $\xi =L/32$ and $\alpha =0.6,0.8,\cdots ,2.0$. The remaining parameters are kept as in Fig. 2.