Criticality of the uniform magnetic order in a ferromagnetic-antiferromagnetic random alternating quantum spin chain

An $S=1∕2$ ferromagnetic (F) and antiferromagnetic (AF) random alternating Heisenberg quantum spin chain model is investigated in connection to its realization compound ${\left(\mathrm{C}{\mathrm{H}}_3\right)}_2\mathrm{C}\mathrm{H}\mathrm{N}{\mathrm{H}}_3\mathrm{Cu}{\left({\mathrm{Cl}}_x{\mathrm{Br}}_{1-x}\right)}_3$. The exchange interaction bonds consist of alternating strong F-AF random bonds and weak uniform AF bonds. Using the quantum Monte Carlo method we have found that the excitation energy gap closes and the uniform AF order becomes critical in the intermediate-concentration region. This finding explains the experimental observation of the magnetic transition by considering weak interchain interactions. The present results suggest that uniform AF order survives even in the presence of randomly located ferromagnetic bonds. This may be an exotic quantum effect.

Figure 1

A theoretical model investigated in this paper. Thick solid and thick dashed lines depict random ferromagnetic and antiferromagnetic interaction bonds. Thin dashed lines depict uniform antiferromagnetic bonds.

Figure 2

(a) Relaxation functions of the staggered magnetization $\left(M_{\mathrm{AF}}\right)$ for various choices of the Trotter number $m$. The inverse temperature $\beta J=50$ for $\beta J∕m=0.10$ and $\beta J=500$ for the other cases. The random bond concentration ratio $p=0.75$. There is no temperature dependence within ${10}^4$ Monte Carlo steps. A real-space system size is 644 spins for $\beta J∕m=0.10$ and 322 spins for the other cases. (b) If the Monte Carlo step is rescaled by $\tau (\beta J∕m)$ of each Trotter number, the relaxation functions fall onto the same curve. The correlation time $\tau \left(0.10\right)=40$, $\tau \left(0.33\right)=4.2$, and $\tau \left(0.50\right)=2.18$.

Figure 3

The excitation energy gap between the subspace with total $S^z=0$ and that with $S^z=1$ plotted versus the ferromagnetic bond concentration $p$. The spin number is denoted by $Ns$. Data denoted by Diag. were obtained by numerical diagonalization, and those denoted by QMC were obtained by quantum Monte Carlo simulations. The error bars are smaller than the symbols.

Figure 4

Nonequilibrium relaxation results of the local susceptibility. The divergence is observed in the gapless phase, $0.025<p<0.75$, where a value of the ferromagnetic bond concentration is denoted beside each plot.

Figure 5

Finite-time scaling results of the local susceptibility on the Haldane side.

Figure 6

Nonequilibrium relaxation results of the string order parameters. A ferromagnetic bond concentration is denoted beside each plot.

Figure 7

A generalized staggered state. Arrows depict the $S^z$ representation. Solid lines depict ferromagnetic bonds and dashed lines depict antiferromagnetic bonds.

Figure 8

Nonequilibrium relaxation of the generalized staggered susceptibility ${\chi }_{g\text{−}\mathrm{AF}}$ and that of the uniform staggered susceptibility ${\chi }_{u\text{−}\mathrm{AF}}$. (a) The dimer side. The data of the ${\chi }_{g\text{−}\mathrm{AF}}$ are multiplied by 2, and those of the ${\chi }_{u\text{−}\mathrm{AF}}$ are divided by 2 in order to separate the plots. (b) The Haldane side. Both orders exhibit the critical behavior in the gapless region $0<p<0.625$.

Figure 9

Nonequilibrium relaxation results of the uniform staggered magnetization $M_{\mathrm{AF}}$. (a) On the dimer side. (b) On the Haldane side. The value of the ferromagnetic bond concentration is denoted beside each plot. The relaxation functions in the pure concentration limits are plotted by circle symbols, and those of the $S=1∕2$ uniform AFH chain are plotted by cross symbols. The relaxation functions on the Haldane side are fitted by the analytical expression $t^{-\alpha ∕z}$, which is obtained by the real-space renormalization procedure.

Figure 10

Depletion procedure of a strong antiferromagnetic bond on the Haldane side.

Figure 11

Nonequilibrium relaxation results of the up-up-down-down staggered magnetization when the simulation starts from the up-up-down-down state. The reference lines are the analytical expressions, $t^{-\alpha ∕z}$, for $M_{\mathrm{AF}}$.

Figure 12

A phase diagram concluded in this paper.