Critical properties of doped coupled spin-Peierls chains

Using numerical Real Space Renormalization Group methods as well as Stochastic Series Expansions Quantum Monte Carlo simulations a generic model of diluted spin-$\frac{1}{2}$ impurities interacting at long distances is investigated. Such a model gives a generic description of coupled dimerized spin-Peierls chains doped with nonmagnetic impurities at temperatures lower than the spin gap. A scaling regime with temperature power-law behaviors in several quantities like the uniform or staggered susceptibilities is identified and interpreted in terms of large clusters of correlated spins.

Figure 1

(Color online) Schematic picture of the doped spin-Peierls system. Thick bonds stand for dimers and nonmagnetic impurities (released spins $\frac{1}{2}$) are represented by open circles (black arrows). The initial RSRG step is illustrated starting from a typical configuration with four impurities: (a) The strongest coupled pair is identified (red line). (b) This pair, e.g., ferromagnetic here, is replaced by a spin $S=1$ (red arrow). (c) The couplings with all other spins (dashed lines) are renormalized.

Figure 2

(Color online) Average effective spin $⟨S^{\mathrm{eff}}⟩$ of the clusters of active spins vs the energy scale $⟨{\Delta }_0⟩$ for six different concentrations $x$ indicated on the plot. Numerical RSRG data obtained for $N_s=1024$ spins over more than ${10}^4$ samples. Inset: for the same samples, average number $⟨n⟩$ of initial spins-$\frac{1}{2}$ per cluster vs $⟨{\Delta }_0⟩$. Dashed lines are power-law fits (see text).

Figure 3

(Color online) Curie constant per spin $⟨C⟩$ plotted vs the energy scale. (a) Quantum Monte Carlo SSE results shown vs $T$ for $N_s=256$ spins and nine different concentrations $x$ indicated on the plot. Disorder is performed on ${10}^3–{10}^4$ samples. (b) Numerical RSRG results shown for $N_s=1024$ spins and six different concentrations $x$ as indicated on the plot, vs the RG energy scale $⟨{\Delta }_0⟩$. Error bars are typically smaller than symbol sizes, the number of samples always exceeding ${10}^4$. The full line correspond to the saturation value of $1∕12$.

Figure 4

(Color online) Direct comparisons between RSRG (full lines) and SSE simulations (symbols) of $⟨C⟩-1∕12$ are shown for the six different concentrations indicated by (a), (b),…, (f) vs the RG energy scale $⟨{\Delta }_0⟩$ or the SSE temperature $T$.

Figure 5

(Color online) Curie constant $⟨C⟩$ plotted vs $T^{{\gamma }_{\mathrm{SSE}}}$ for the QMC data in (a) and vs ${⟨{\Delta }_0⟩}^{{\gamma }_{\mathrm{RG}}}$ for the RSRG data in (b), in order to obtain the best data collapse at low energy.

Figure 6

(Color online) $T⟨{\chi }_{\text{stag}}\left(T\right)⟩$ plotted vs $T$ for five different concentrations. Full lines are fits corresponding to power-law behaviors $\sim T^{-2{\gamma }^{\prime }}$. All data are computed by QMC using $N_s=256$ random spins and averaged over disorder.

Figure 7

(Color online) Exponents $\alpha \left(x\right)$, $\gamma \left(x\right)$, and ${\gamma }^{\prime }\left(x\right)$ extracted from various SSE and RSRG data as indicated on the plot (see text for details). Straight lines are $\sim \sqrt{x}$.