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arXiv:1912.10788 · Physical review. B./Physical review. B

Piercing the rainbow state: Entanglement on an inhomogeneous spin chain with a defect

Nadir Samos Sáenz de Buruaga, Silvia N. Santalla, Javier Rodríguez-Laguna, Germán Sierra

Published May 15, 2020• 14 citations
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Abstract

The rainbow state denotes a set of valence bond states organized concentrically around the center of a spin 1/2 chain. It is the ground state of an inhomogeneous XX Hamiltonian and presents a maximal violation of the area law of entanglement entropy. Here, we add a tunable exchange coupling constant at the center, $\ensuremath{\gamma}$, and show that it induces entanglement transitions of the ground state. At very strong inhomogeneity, the rainbow state survives for $0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}1$, while outside that region the ground state is a product of dimers. In the weak inhomogeneity regime, the entanglement entropy satisfies a volume law, derived from CFT in curved space-time, with an effective central charge that depends on the inhomogeneity parameter and $\ensuremath{\gamma}$. In all regimes we have found that the entanglement properties are invariant under the transformation $\ensuremath{\gamma}\ensuremath{\longleftrightarrow}1\ensuremath{-}\ensuremath{\gamma}$, whose fixed point $\ensuremath{\gamma}=1/2$ corresponds to the usual rainbow model. Finally, we study the robustness of nontrivial topological phases in the presence of the defect.

DOI
10.1103/physrevb.101.205121

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The rainbow state denotes a set of valence bond states organized concentrically around the center of a spin 1/2 chain. It is the ground state of an inhomogeneous XX Hamiltonian and presents a maximal violation of the area law of entanglement entropy. Here, we add a tunable exchange coupling constant at the center, \ensuremathγ\ensuremath{\gamma}\ensuremathγ, and show that it induces entanglement transitions of the ground state. At very strong inhomogeneity, the rainbow state survives for 0\ensuremath≤\ensuremathγ\ensuremath≤10\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}10\ensuremath≤\ensuremathγ\ensuremath≤1, while outside that region the ground state is a product of dimers. In the weak inhomogeneity regime, the entanglement entropy satisfies a volume law, derived from CFT in curved space-time, with an effective central charge that depends on the inhomogeneity parameter and \ensuremathγ\ensuremath{\gamma}\ensuremathγ. In all regimes we have found that the entanglement properties are invariant under the transformation \ensuremathγ\ensuremath⟷1\ensuremath−\ensuremathγ\ensuremath{\gamma}\ensuremath{\longleftrightarrow}1\ensuremath{-}\ensuremath{\gamma}\ensuremathγ\ensuremath⟷1\ensuremath−\ensuremathγ, whose fixed point \ensuremathγ=1/2\ensuremath{\gamma}=1/2\ensuremathγ=1/2 corresponds to the usual rainbow model. Finally, we study the robustness of nontrivial topological phases in the presence of the defect.
An exponential deformation of a 1D critical Hamiltonian, with couplings falling on a length scale h−1h^{-1}h−1, gives rise to ground states whose entanglement entropy follows a volume law, i.e. the area law is violated maximally. The ground state is now in the so-called {\em rainbow phase}, where valence bonds connect sites on the left half with their symmetric counterparts on the right. Here we discuss some of the most relevant features of this rainbow phase, focusing on the XX and Heisenberg models. Moreover, we show that the rainbow state can be understood either as a thermo-field double of a conformal field theory with a temperature proportional to hhh or as a massless Dirac fermion in a curved spacetime with constant negative curvature proportional to hhh. Finally, we introduce a study of the time-evolution of the rainbow state after a quench to a homogeneous Hamiltonian.
Entanglement can be quantified in bipartite systems using the entanglement entropy which is, in many cases, bounded by an area law. Nonetheless, some critical 1D systems violate this area law and present logarithmic corrections which are parametrized by the central charge of the associated conformal field theory (CFT). Furthermore, other violations to the area law may appear in random systems and inhomogeneous systems. In the first part of the work we focus on the properties of entanglement in random hopping models, focusing on the similarities between the CFT predictions for the clean case and the strong disorder renormalization group (SDRG) predictions. We use a combination of methods: exact diagonalization, a transformation of the SDRG and a new tool based on the study of random permutations. All techniques coincide in providing a compelling image, based on a bond-picture. The second part focus on the engineering of an inhomogeneous fermionic system, in order to obtain a maximal growth of the block entanglement entropy in the ground state. We show that the deformation parameter allows us to interpolate between an uniform model and the strong disorder limit, which is a maximally entangled state called the rainbow state. This state can be seen as a thermo field state. The last part focus on time-evolution of some valence bond states after a global quench to a homogeneous Hamiltonian in 1D. We focused on the evolution of the entanglement in the rainbow and dimer states. We proposed a generalization of the quasiparticle picture, which we have called the ballistic picture.
In one dimension the area law for the entanglement entropy is violated maximally by the ground state of a strongly inhomogeneous spin chain, the so called concentric singlet phase (CSP), that looks like a rainbow connecting the two halves of the chain. In this paper we show that, in the weak inhomogeneity limit, the rainbow state is a thermo field double of a conformal field theory with a temperature proportional to the inhomogeneity parameter. This result suggests some relation of the CSP with black holes. Finally, we propose an extension of the model to higher dimensions.
The entanglement entropy of the ground state of a quantum lattice model with local interactions usually satisfies an area law. However, in 1D systems some violations may appear in inhomogeneous systems or in random systems. In our inhomogeneous system, the inhomogeneity parameter, h, allows us to tune different regimes where a volumetric violation of the area law appears. We apply the strong disorder renormalization group to describe the maximally entangled state of the system in a strong inhomogeneity regime. Moreover, in a weak inhomogeneity regime, we use a continuum approximation to describe the state as a thermo-field double in a conformal field theory with an effective temperature which is proportional to the inhomogeneity parameter of the system. The latter description also shows that the universal scaling features of this model are captured by a massless Dirac fermion in a curved space-time with constant negative curvature R = h2, providing another example of the relation between quantum entanglement and space-time geometry. The results we discuss here were already published before, but here we present a more didactic exposure of basic concepts of the rainbow system for the students attending the Latin American School of Physics “Marcos Moshinsky” 2017.
We study an inhomogeneous critical Ising chain in a transverse field whose couplings decay exponentially from the center. In the strong inhomogeneity limit, we apply Fisher's renormalization group to show that the ground state is formed by concentric singlets similar to those of the rainbow state of the XX model. In the weak inhomogeneity limit, we map the model to a massless Majorana fermion living in a hyperbolic space-time, where, using conformal field theory techniques, we derive the entanglement entropy and show that it presents a linear scaling. We also study an inhomogeneous noncritical Ising model that for weak inhomogeneity is mapped to a massive Majorana fermion, while for strong inhomogeneity regime, it exhibits trivial and nontrivial topological phases and a separation between regions with high and low entanglement. We also present the entanglement Hamiltonian of the model.
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.