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arXiv:1812.10023 · Journal of Physics A Mathematical and Theoretical

Holographic duals of inhomogeneous systems: the rainbow chain and the sine-square deformation model

Ian MacCormack, Aike Liu, Masahiro Nozaki, Shinsei Ryu

Published Aug 7, 2019• 48 citations
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Abstract

Abstract A holographic dual description of inhomogeneous systems is discussed. Notably, finite temperature results for the entanglement entropy in both the rainbow chain and the sine-square deformation model are obtained holographically by choosing appropriate foliations of the Bañados– Teitelboim–Zanelli spacetime. Other inhomogeneous theories are also discussed. The entanglement entropy results are verified numerically, indicating that a wide variety of inhomogeneous field theory phenomenology can be seen in different slicings of asymptotically AdS 3 spacetimes.

DOI
10.1088/1751-8121/ab3944

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•Aug 15, 2016•30 citations
Abstract A holographic dual description of inhomogeneous systems is discussed. Notably, finite temperature results for the entanglement entropy in both the rainbow chain and the sine-square deformation model are obtained holographically by choosing appropriate foliations of the Bañados– Teitelboim–Zanelli spacetime. Other inhomogeneous theories are also discussed. The entanglement entropy results are verified numerically, indicating that a wide variety of inhomogeneous field theory phenomenology can be seen in different slicings of asymptotically AdS 3 spacetimes.
Abstract The rainbow chain is an inhomogenous exactly solvable local spin model that, in its ground state, displays a half-chain entanglement entropy growing linearly with the system size. Although many exact results about the rainbow chain are known, the structure of the underlying quantum field theory has not yet been unraveled. Here we show 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 = − h 2 ( h is the amplitude of the inhomogeneity). This identification allows us to use recently developed techniques to study inhomogeneous conformal systems and to analytically characterise the entanglement entropies of more general bipartitions. These results are carefully tested against exact numerical calculations. Finally, we study the entanglement entropies of the rainbow chain in thermal states, and find that there is a non-trivial interplay between the rainbow effective temperature T R and the physical temperature T .
In recent years, the holographic duality between T T ¯ -deformed conformal field theory (CFT) and Anti-de Sitter (AdS) spacetime with finite radial cutoff has received significant attention. The study of T T ¯ deformation within the framework of de Sitter (dS)/CFT duality has also progressed. This paper shows that the trace flow equation in dS spacetime can be analytically extended from its AdS counterpart through double Wick rotations. Meanwhile, we generalize the replica method in both AdS and dS holography to derive a general expression for the entanglement entropy of arbitrary single spatial intervals within the T T ¯ -deformed framework. For both finite size and finite temperature systems, we obtain the analytical expression for the entanglement entropy after T T ¯ deformation. Finally, in dS/dS holography and half-dS holography, we find that the dual field theory exhibits nonlocality by analyzing the strong subadditivity and boosted strong subadditivity of entanglement entropy.
The rainbow chain is an inhomogenous exactly solvable local spin model that, in its ground state, displays a half-chain entanglement entropy growing linearly with the system size. Although many exact results about the rainbow chain are known, the structure of the underlying quantum field theory has not yet been unraveled. Here we show 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=−h2R=-h^2R=−h2 (hhh is the amplitude of the inhomogeneity). This identification allows us to use recently developed techniques to study inhomogeneous conformal systems and to analytically characterise the entanglement entropies of more general bipartitions. These results are carefully tested against exact numerical calculations. Finally, we study the entanglement entropies of the rainbow chain in thermal states, and find that there is a non-trivial interplay between the rainbow effective temperature TRT_RTR​ and the physical temperature TTT.
Entanglement entropy (EE) is a concept of paramount importance in modern theoretical physics. Nevertheless, calculating EE in generic interacting theories with non-trivial time dependence is notoriously difficult. In this thesis, we address the real-time dynamics of EE in certain strongly-coupled quantum field theories (QFT) using the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence. We first study the time evolution of EE in an exactly solvable model of two-dimensional CFT with an inhomogeneous sine-square deformation (SSD). We argue that the non-trivial time evolutions driven by the SSD Hamiltonian can be regarded as changing the foliations of spacetime. Remarkably, this leads to the presence of new horizons in the dual three-dimensional bulk. We then compute the EE in SSD CFT using holography and found agreement with CFT results in all comparable cases. Afterwards, we proceed to entanglement dynamics in generic chaotic quantum many-body systems. In the limit of large subsystems and late times, entanglement dynamics is believed to be described by an entanglement membrane effective theory. For 2d CFTs, a degenerate tension function reproduces several results regarding the dynamics of the entropy; this stands in contrast to higher dimensions where the tension is non-degenerate. We use holography to show that, in order to correctly capture the reflected entropy in 2d CFT, one needs to add an additional degree of freedom to the membrane description. Furthermore, we show that the conventional non-degenerate membrane tension function emerges upon introducing a relevant deformation to the CFT, dual to a planar Banados-Teitelboim-Zanelli (BTZ) black hole with scalar hair and with an interior Kasner universe. Subsequently, we establish a membrane effective theory for the dynamics of reflected entropy in higher dimensions. Lastly, we apply the membrane theories for reflected entropy to study the time evolution of multiparty entanglement signals. We found that scrambling dynamics generate multiparty entanglement with rich structures.
Abstract Inhomogeneous quantum critical systems in one spatial dimension have been studied by using conformal field theory in static curved backgrounds. Two interesting examples are the free fermion gas in the harmonic trap and the inhomogeneous XX spin chain called rainbow chain. For conformal field theories defined on static curved spacetimes characterised by a metric which is Weyl equivalent to the flat metric, with the Weyl factor depending only on the spatial coordinate, we study the entanglement hamiltonian and the entanglement spectrum of an interval adjacent to the boundary of a segment where the same boundary condition is imposed at the endpoints. A contour function for the entanglement entropies corresponding to this configuration is also considered, being closely related to the entanglement hamiltonian. The analytic expressions obtained by considering the curved spacetime which characterises the rainbow model have been checked against numerical data for the rainbow chain, finding an excellent agreement.
We compute holographic entanglement entropy in two strongly coupled nonlocal field theories: the dipole and the noncommutative deformations of SYM theory. We find that entanglement entropy in the dipole theory follows a volume law for regions smaller than the length scale of nonlocality and has a smooth cross-over to an area law for larger regions. In contrast, in the noncommutative theory the entanglement entropy follows a volume law for up to a critical length scale at which a phase transition to an area law occurs. The critical length scale increases as the UV cutoff is raised, which is indicative of UV/IR mixing and implies that entanglement entropy in the noncommutative theory follows a volume law for arbitrary large regions when the size of the region is fixed as the UV cutoff is removed to infinity. Comparison of behaviour between these two theories allows us to explain the origin of the volume law. Since our holographic duals are not asymptotically AdS, minimal area surfaces used to compute holographic entanglement entropy have novel behaviours near the boundary of the dual spacetime. We discuss implications of our results on the scrambling (thermalization) behaviour of these nonlocal field theories. ar X iv