Strong Disorder Renormalization Group Method for Bond Disordered Antiferromagnetic Quantum Spin Chains with Long Range Interactions: Ground State Properties

Abstract: We introduce and implement a reformulation of the strong disorder renormalization group method in real space, well suited to study bond disordered antiferromagnetic power law coupled quantum spin chains. We derive the Master equations for the distribution function of pair distances $\tilde{r}$. First, we apply it to a short range coupled spin chain, keeping only interactions for adjacent spins. We confirm that it is solved by the infinite randomness fixed point distribution. Then, we solve the Master equation for the power law long range interaction between all spins for any anisotropy ranging from the XX-limit to the isotropic Heisenberg limit, corresponding to a tight binding chain of disordered long range interacting Fermions with long range hopping. We thereby show that the distribution function of couplings $J$ at renormalization scale $\Omega$ flows to the strong disorder fixed point distribution with small corrections at $\tilde{r} > \rho,$ which depend on power exponent $\alpha$ and coupling anisotropy $\gamma.$ As a consequence, the low temperature magnetic susceptibility diverges with an anomalous power law. The distribution of singlet lengths $l$ is found to decay as $l^{-2}$. The entanglement entropy of a subsystem of length $n$ increases in the ground state logarithmically for all $\alpha$ and $\gamma$. After a global quantum quench the entanglement entropy increases with time logarithmically as $S(t) \sim \ln(t)/(2\alpha)$.