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On entanglement dynamics in holographic theories

Hanzhi Jiang

Published Jan 1, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.5287/ora-mznqvwepw

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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.
A bstract The time evolution of entanglement entropy in generic chaotic many-body systems has an effective description in terms of a minimal membrane, characterised by a tension function. 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. In this paper 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 of the CFT, dual to a planar BTZ black hole with scalar hair and with an interior Kasner universe. Finally, we also study the membrane description for reflected entropy and information velocity [1] in higher dimensions.
A holographic correspondence has been recently developed between higher spin theories in three-dimensional anti-de Sitter space (AdS3) and two-dimensional Conformal Field Theories (CFTs) with extended symmetries. A class of such dualities involves SL(N, R) × SL(N, R) Chern-Simons gauge theories in the (2 + 1)-dimensional bulk space-time, and CFTs with WN symmetry algebras on the (1 + 1)-dimensional boundary. The topological character of the bulk theory forces one to reconsider standard geometric notions such as black hole horizons and entropy, as well as the usual holographic dictionary. Motivated by this challenge, in this note we present a proposal to compute entanglement entropy in the WN CFTs via holographic methods. In particular, we introduce a functional constructed from Wilson lines in the bulk Chern-Simons theory that captures the entanglement entropy in the CFTs dual to standard AdS3 gravity, corresponding to SL(2, R) × SL(2, R) gauge group, and admits an immediate generalization to the higher spin case. We explicitly evaluate this functional for several known solutions of the bulk theory, including charged black holes dual to thermal CFT states carrying higher spin charge, and show that it reproduces expected features of entanglement entropy, study whether it obeys strong subadditivity, and moreover show that it reduces to the thermal entropy in the appropriate limit.
Time-dependent entanglement entropy (EE) is computed for a single interval in two-dimensional conformal theories from a quenched initial state in the presence of spatial boundaries. The EE is found to be periodic in time with periodicity equal to the system size LLL. For large enough LLL, the EE shows a rise to a thermal value (characterized by a temperature 1/β1/β1/β determined by the initial state), followed by periodic returns to the original value. This works irrespective of whether the conformal field theory (CFT) is rational or irrational. For conformal field theories with a holographic dual, the large ccc limit plays an essential role in ensuring that the EE computed from the CFT is universal (independent of the details of the CFT and of boundary conditions) and is exactly matched by the holographic EE. The dual geometry is computed and it interpolates between a BTZ black hole at large LLL and global AdS at large βββ.
We further develop perturbative methods used to calculate entanglement entropy (EE) away from an interacting CFT fixed point. At second order we find certain universal terms in the renormalized EE which were predicted previously from holography and which we find hold universally for relevant deformations of any CFT in any dimension. We use both replica methods and direct methods to calculate the EE and in both cases find a non-local integral expression involving the CFT two point function. We show that this integral expression can be written as a local integral over a higher dimensional bulk modular hamiltonian in an emergent AdS space-time. This bulk modular hamiltonian is associated to an emergent scalar field dual to the deforming operator. We generalize to arbitrary spatially dependent couplings where a linearized metric emerges naturally as a way of efficiently encoding the field theory entanglement: by demanding that Einstein’s equations coupled to the bulk scalar field are satisfied, we show that EE can be calculated as the area of this metric. Not only does this show a direct emergence of a higher dimensional gravitational theory from any CFT, it allows for effective evaluation of the the integrals required to calculate EE perturbativly. Our results can also be interpreted as relating the non-locality of the modular hamiltonian for a spherical region in non-CFTs and the non-locality of the holographic bulk to boundary map.
The determination of structures that can emerge out of the interactions among the constituents of a quantum many-body system is a foundational task of condensed matter physics.One of the most advanced proposals within this paradigm is the holographic generation of spacetime in some strongly coupled chaotic systems with particular patterns of entanglement and quantum complexity, which is historically motivated by the holographic principle and by the AdS/CFT correspondence.As a part of this scenario, this thesis is dedicated to the analysis of some concepts that are relevant to such proposal, where they are separately applied to some lattice models.Concretely, the matter is divided into three main studies: first, the calculation of the time-dependent circuit complexity in the Ising model with a periodically driven transverse field, where we establish the effectiveness of this quantity for the detection of nonequilibrium quantum phase transitions.Our results provide hints for understanding how universal features out of equilibrium are captured by the complexity of quantum states.Second, the derivation of a bound on the maximal rate of entanglement entropy production for a class of one-dimensional quantum circuits with periodic dynamics.An example of a circuit that saturates the bound is composed by parallel SWAP gates acting on entangled pairs.Out of inequalities obeyed by the entropy, in addition to considerations on multipartite entanglement, we indicate that the effect of a chaotic dynamics cannot result in the increase on the rate of entanglement production.Third, the construction of a class of supersymmetric many-body systems using symmetric inverse semigroups.For particular toy models built out of this structure, we study some questions regarding supersymmetric phases, integrability, disorder, spreading of quantum information and many-body localization.Finally, besides those three works, we include an essay addressing the problem of emergent holographic spaces out of two-dimensional conformal field theories, where the mediation between the two parts is performed by means of a Riemannian structure defined in the Hilbert space of the field theory.
This thesis is dedicated to the analysis of correlation structure in two quite different ex-amples. In the first one we investigate the entanglement entropy (EE) through conformal inter-faces in two-dimensional conformal field theories. The EE is a measure for the strength of entanglement between two subsystems of a system in a pure state. Interfaces are one-dimensional objects on a two-dimensional surface along which two conformal field theories are glued together. They are called conformal if their gluing conditions is invariant under conformal transformations that preserve the shape of the interface. Special interfaces are topological ones. They are called topological because the partition function does not change when the interface is continuously deformed. We in particular show that in a vast class of conformal field theories the presence of a topological interface at the boundary between two subregions makes the EE get dressed with an additional sub-leading but universal contribution that solely depends on the interface data. This sub-leading contribution can be interpreted as a relative entropy measuring the relative loss of entanglement compared to no interface insertion. We also compute the EE through general conformal interfaces in the critical Ising model and find that its leading order is affected by the interfaces if and only if it is not topological. Any conformal defect can be characterized by its transmissivity. As physically expected, the EE though conformal interfaces in the Ising model decreases for lower transmissivities. The second example is what we call classical holographic codes. They are classical probabilistic codes defined by a network on a uniform tiling of a constant time slice of AdS 3 -spacetime. They share some remarkable properties with certain quantum error correcting codes that are designed to mimic particular properties of the AdS/CFT correspondence. Under these features are the Ryu-Takayanagi formula and bulk reconstruction properties that both reflect deep connections between the correlation structure of a theory and the ge- ometry of its dual description. Our classical codes can be seen as toy models for holography that show that the latter features do not necessarily originate from a quantum description.