Unusual area-law violation in random inhomogeneous systems

TL;DR

This paper uncovers a unique area law violation in a disordered inhomogeneous spin chain, revealing a power-law entanglement growth with subsystem size, and explores the effects of interactions on this phenomenon.

Contribution

It demonstrates a novel area law violation in a disordered spin chain and links the entanglement scaling to random walk probabilities, extending understanding of entanglement in disordered systems.

Findings

Entanglement entropy exhibits power-law growth with subsystem size (exponent 1/2).

Area law violation occurs only at the free-fermion point in the phase diagram.

Numerical evidence supports SDRG predictions using free-fermion solutions.

Abstract

The discovery of novel entanglement patterns in quantum many-body systems is a prominent research direction in contemporary physics. Here we provide the example of a spin chain with random and inhomogeneous couplings that in the ground state exhibits a very unusual area law violation. In the clean limit, i.e., without disorder, the model is the rainbow chain and has volume law entanglement. We show that, in the presence of disorder, the entanglement entropy exhibits a power-law growth with the subsystem size, with an exponent 1/2. By employing the Strong Disorder Renormalization Group (SDRG) framework, we show that this exponent is related to the survival probability of certain random walks. The ground state of the model exhibits extended regions of short-range singlets (that we term ``bubble'' regions) as well as rare long range singlet (``rainbow'' regions). Crucially, while the probability of extended rainbow regions decays exponentially with their size, that of the bubble regions is power law. We provide strong numerical evidence for the correctness of SDRG results by exploiting the free-fermion solution of the model. Finally, we investigate the role of interactions by considering the random inhomogeneous XXZ spin chain. Within the SDRG framework and in the strong inhomogeneous limit, we show that the above area-law violation takes place only at the free-fermion point of phase diagram. This point divides two extended regions, which exhibit volume-law and area-law entanglement, respectively.