More on the rainbow chain: entanglement, space-time geometry and thermal states

Rodríguez-Laguna, Javier; Dubail, Jérôme; Ramírez, Giovanni; Calabrese, Pasquale; Sierra, Germȧn

doi:10.1088/1751-8121/aa6268

Cited by 50 publications

(87 citation statements)

References 68 publications

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“…through the method of the cutoff extraction Eq. (34). We focus on the Von Neumann entropy S1 and on the first Rényi S2 for ∆ = ±0.5 and several magnetic trap (see insets).…”

Section: Fig 3 Comparison Of Entanglement Entropies In the Xxz Spinmentioning

confidence: 99%

Entanglement entropies of inhomogeneous Luttinger liquids

Bastianello

Dubail

Stéphan

2020

J. Phys. A: Math. Theor.

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We develop a general framework to compute the scaling of entanglement entropy in inhomogeneous one-dimensional quantum systems belonging to the Luttinger liquid universality class. While much insight has been gained in homogeneous systems by making use of conformal field theory techniques, our focus is on systems for which the Luttinger parameter K depends on position, and conformal invariance is broken. An important point of our analysis is that contributions stemming from the UV cutoff have to be treated very carefully, since they now depend on position. We show that such terms can be removed either by considering regularized entropies specifically designed to do so, or by tabulating numerically the cutoff, and reconstructing its contribution to the entropy through the local density approximation. We check our method numerically in the spin-1/2 XXZ spin chain in a spatially varying magnetic field, and find excellent agreement. arXiv:1910.09967v1 [cond-mat.stat-mech]

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“…through the method of the cutoff extraction Eq. (34). We focus on the Von Neumann entropy S1 and on the first Rényi S2 for ∆ = ±0.5 and several magnetic trap (see insets).…”

Section: Fig 3 Comparison Of Entanglement Entropies In the Xxz Spinmentioning

confidence: 99%

Entanglement entropies of inhomogeneous Luttinger liquids

Bastianello

Dubail

Stéphan

2020

J. Phys. A: Math. Theor.

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“…In the spin representation, the state (8) corresponds to a product of singlets between the sites (−n, +n) of the chain. An important feature of the rainbow state (8) is that the entanglement entropy of a subsystem starting from the chain center grows linearly with its size (corresponding to a volume law) as [1,[62][63][64][65][66] lim h→∞ S(h, ) = ln 2,…”

Section: The Random Inhomogeneous XX Chain (Randbow Chain)mentioning

confidence: 99%

Unusual area-law violation in random inhomogeneous systems

Alba¹,

Santalla²,

Ruggiero³

et al. 2019

J. Stat. Mech.

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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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“…Although the so-called 'Rainbow spin chain' is not random, the spatial structure of its inhomogeneity allows to apply iteratively the Ma-Dasgupta SDRG rule to construct the ground state and analyze its entanglement properties [89][90][91][92][93]. The addition of disorder in this rainbow spin chain has been also studied recently via SDRG [94].…”

Section: F the Rainbow Spin Chainmentioning

confidence: 99%

Strong disorder RG approach – a short review of recent developments

Iglói

Monthus

2018

Eur. Phys. J. B

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Relations between SDRG and Entanglement-Algorithms 10 VI. Localized and Many-Body-Localized Phases of quantum spin chains 11 A. RSRG-X for excited eigenstates 11 B. RSRG-t for the unitary dynamics 11 C. Non-equilibrium dynamical scaling of observables 12 D. Comparison with other RG procedures existing in the field of Many-Body-Localization 13 VII. Floquet dynamics of periodically driven chains in their localized phases 13 VIII. Open dissipative quantum spin chains 13 A. Quantum spin chains coupled to a bath of quantum oscillators 13 B. Lindblad dynamics for random quantum spin chains 14 IX. Anderson localization models 14 X. Random contact process 14 A. Strong disorder RG rules 15 B. Long range spreading 15 C. Temporal disorder 15 XI. Classical master equations 16 A. Real-space renormalization for random walks in random media 16 B. RG in configuration-space for the dynamics of classical many-body models 16 XII. Random classical oscillators 17 A. Random elastic networks 17 B. Synchronisation of interacting non-linear dissipative classical oscillators 17 XIII. Other classical models 17 A. Equilibrium properties of random systems 17 B. Extremes of stochastic processes 18 XIV. Conclusion 18

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More on the rainbow chain: entanglement, space-time geometry and thermal states

Cited by 50 publications

References 68 publications

Entanglement entropies of inhomogeneous Luttinger liquids

Entanglement entropies of inhomogeneous Luttinger liquids

Unusual area-law violation in random inhomogeneous systems

Strong disorder RG approach – a short review of recent developments

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