Entanglement Properties of Disordered Quantum Spin Chains with Long-Range Antiferromagnetic Interactions

TL;DR

This paper investigates entanglement properties in disordered quantum spin chains with long-range interactions, revealing critical behavior with a universal entanglement entropy and power-law decay of concurrence through analytical and numerical methods.

Contribution

It introduces a correction scheme to the strong disorder renormalization group approach, enabling accurate analysis of typical concurrence and entanglement entropy in long-range disordered spin chains.

Findings

Entanglement entropy exhibits logarithmic growth with a universal central charge c=ln 2.

Concurrence shows power-law decay with distance, with exponents depending on the method used.

The system displays critical behavior similar to the Haldane-Shastry model across a range of interaction decay exponents.

Abstract

We examine the concurrence and entanglement entropy in quantum spin chains with random long-range couplings, spatially decaying with a power-law exponent $\alpha$. Using the strong disorder renormalization group (SDRG) technique, we find by analytical solution of the master equation a strong disorder fixed point, characterized by a fixed point distribution of the couplings with a finite dynamical exponent, which describes the system consistently in the regime $\alpha > 1/2$. A numerical implementation of the SDRG method yields a power law spatial decay of the average concurrence, which is also confirmed by exact numerical diagonalization. However, we find that the lowest-order SDRG approach is not sufficient to obtain the typical value of the concurrence. We therefore implement a correction scheme which allows us to obtain the leading order corrections to the random singlet state. This approach yields a power-law spatial decay of the typical value of the concurrence, which we derive both by a numerical implementation of the corrections and by analytics. Next, using numerical SDRG, the entanglement entropy (EE) is found to be logarithmically enhanced for all $\alpha$, corresponding to a critical behavior with an effective central charge $c = {\rm ln} 2$, independent of $\alpha$. This is confirmed by an analytical derivation. Using numerical exact diagonalization (ED), we confirm the logarithmic enhancement of the EE and a weak dependence on $\alpha$. For a wide range of distances $l$, the EE fits a critical behavior with a central charge close to $c=1$, which is the same as for the clean Haldane-Shastry model with a power-la-decaying interaction with $\alpha =2$. Consistent with this observation, we find using ED that the concurrence shows power law decay, albeit with smaller power exponents than obtained by SDRG.