Unusual area-law violation in random inhomogeneous systems

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. arXiv:1807.04179v1 [cond-mat.str-el] A FIG. 1: Setup used in this work. (top) Definition of the random inhomogeneous XX chain.The chain couplings are denoted as Jn = e −|n|h Kn, with n half integer numbers, h a real inhomogeneity parameter, and Kn independent random variables distributed with (4). In this work we focus on the entanglement entropy of a subregion A of length (shaded area in the Figure). Subsystem A starts from the chain center. local, translational invariant models that exhibit area-law violations has been constructed by Movassagh and Shor in Ref. [44]. Their ground-state entanglement entropy is ∝ 1/2 , thus exhibiting a polynomial violation of the area law. Importantly, the exponent of the entanglement growth originates from universal properties of the random walk. This is due to the fact that the ground state of the model is written in terms of a special class of combinatorial objects, called Motzkin paths [51]. A similar result can be obtained [45] using the Fredkin gates [52]. An interesting generalization of Ref. 44 obtained by deforming a colored version of the Motzkin paths has been presented in Ref. 46. The ground-state phase diagram of the model exhibits two phases with area-law and volume-law entanglement, respectively. These are separated by a "special" point, where the ground state displays square-root entanglement scaling.In this paper, we show that unusual area-law violations can be obtained in a one-dimensional inhomogeneous local system in the presence of disorder. Specifically, here we investiga...

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