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 $Ω$ flows to the strong disorder fixed point distribution with small corrections at $\tilde{r} > ρ,$ which depend on power exponent $α$ and coupling anisotropy $γ.$ 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 $α$ and $γ$. After a global quantum quench the entanglement entropy increases with time logarithmically as $S(t) \sim \ln(t)/(2α)$.

Figures (9)

• Figure 1: SDRG step in real space for a chain of randomly placed spins (circles): The decimation of the strongest coupled spin pair $(i,j)$ (shaded), whose coupling defines the RG scale $\Omega,$ is followed by the renormalization of all positions of spins, ${\bf r}_{l} \rightarrow \tilde{{\bf r}}_{l}$ as the RG scale $\Omega - d\Omega$ is reduced.
• Figure 2: Renormalized anisotropy $\tilde{\gamma}$ as function of bare anisotropy $\gamma.$
• Figure 3: Strong disorder RG step for bond disordered short range coupled spin chains: Decimation of strongest coupled spin pair $(i,j)$, highlighted by the shaded area, whose coupling defines the RG scale $\Omega.$ It is followed by renormalization of the positions of spins, ${\bf r}_{l} \rightarrow \tilde{{\bf r}}_{l}$ and a reduction of the RG scale to $\Omega - d\Omega.$
• Figure 4: The renormalization function Eq. (\ref{['rgelr']}) as function of distances $R_L,R_R$ in units of $\rho,$ for $\alpha = 100,50,10,5,1, 0.5$ from the bottom up, as bounded by $r-2\rho= R_L+R_R-\rho$ from below and $r=R_L+R_R+\rho$ from above.
• Figure 5: Line plot of the correction term to the Master equation as function of $\tilde{r},$ for various values of power $\alpha,$ for $\gamma=0$.