Strong Disorder Renormalization Group Method for Bond Disordered Antiferromagnetic Quantum Spin Chains with Long Range Interactions: Excited States and Finite Temperature Properties

TL;DR

This work extends the real-space SDRG-X framework to excited states and finite-temperature properties of bond-disordered antiferromagnetic spin chains with long-range power-law interactions. For short-range couplings, the excited-state spectrum preserves the infinite randomness fixed point (IRFP) distribution, while long-range interactions introduce finite-width corrections that depend on the exponent $oldsymbol{ extalpha}$; a finite-temperature Master equation yields the temperature dependence of the coupling distributions and related observables. The authors derive thermodynamic and dynamical properties, including magnetic susceptibility, entanglement entropy, spin correlations, and quench-induced entanglement growth, highlighting the role of paramagnetic $S=1/2$ and $S=1$ excitations and the emergence of rainbow-like behavior for $oldsymbol{ extalpha}<2$. They identify regimes where the SDRG is still valid ($oldsymbol{ extalpha}oldsymbol{>oldsymbol{2}}$) and regimes where new renormalization terms (three-point and pseudospin interactions) become important, pointing to future work on multi-spin SDRG extensions and potential higher-dimensional generalizations.

Abstract

We extend the recently introduced strong disorder renormalization group method in real space, well suited to study bond disordered antiferromagnetic power law coupled quantum spin chains, to study excited states, and finite temperature properties. First, we apply it to a short range coupled spin chain, which is defined by the model with power law interaction, keeping only interactions between adjacent spins. We show that the distribution of the absolute value of the couplings is the infinite randomness fixed point distribution. However, the sign of the couplings becomes distributed, and the number of negative couplings increases with temperature $T.$ Next, we derive the Master equation for the power law long range interaction between all spins with power exponent $α$. While the sign of the couplings is found to be distributed, the distribution of the coupling amplitude is given by the strong disorder distribution with finite width $2α,$ with small corrections for $α>2$. Resulting finite temperature properties of both short and power law long ranged spin systems are derived, including the magnetic susceptibility, concurrence and entanglement entropy.

Figures (11)

• Figure 1: 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 2: Schematic SDRG-X procedure: at each RG step four possible pair states are indicated by blue, black and red lines ( with pair energies $E=-J/2,0,0,+J/2$, respectively). After $N/2$ RG steps the many body eigenstates with total energy E is obtained, following a specific SDRG path.
• Figure 3: Strong disorder RG step for bond disordered long 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 $\Omega - d\Omega.$ The initial couplings are indicated by the blue dashed lines.
• Figure 4: Renormalized couplings $\tilde{J}[s]$ are plotted for different projection states $s = 0$ (upper), $s = 1,2$ (middle) and $s = 3$ (lower) in units of RG scale $\Omega$ as function of distances $R_L=r_{li}$ and $R_R=r_{jm}$ in units of RG distance $\rho = (\Omega/\Omega_0)^{-1/\alpha}$ for $\alpha=2$.
• Figure 5: Same as Fig. \ref{['RGalpha2']} but for $\alpha=1/2$.