Entanglement entropy and conformal field theory

By Pasquale Calabrese, J. Cardy. 2009

1,734 citations recorded by the source corpus.

We review the conformal field theory approach to entanglement entropy in 1+1 dimensions. We show how to apply these methods to the calculation of the entanglement entropy of a single interval, and the generalization to different situations such as finite size, systems with boundaries and the case of several disjoint intervals. We discuss the behaviour away from the critical point and the spectrum of the reduced density matrix. Quantum quenches, as paradigms of non-equilibrium situations, are also considered.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/240b723b5e6a5fe24f707d6fadbf65c07db0d4f4

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In Artificial Intelligence

1. A. Nahum
2. T. Takayanagi
3. R. Myers
4. Matthew Fisher
5. V. Khemani
6. Pasquale Calabrese
7. Thomas Hartman
8. T. Nishioka

In Atomic and Molecular Physics, and Optics

1. R. Myers
2. A. Nahum
3. T. Takayanagi
4. V. Khemani
5. Thomas Hartman
6. T. Nishioka
7. T. Faulkner
8. Pasquale Calabrese

Citations in the corpus

• Islands in cosmology (2020, neutral): “3) z2 = rI − tI + iβ/4 z̄2 = rI + tI − iβ/4 z3 = −rR − tR − iβ/4 z̄3 = −rR + tR + iβ/4 z4 = rR − tR − iβ/4 z̄4 = rR + tR + iβ/4 We will calculate the Renyi partition function Zn = Tr (ρI∪R) n using the twist operator methods of Cardy and Calabrese [100, 101], as applied to this problem in [53] (see ”
• Colloquium : Many-body localization, thermalization, and entanglement (2018, background): “The logarithmic propagation of entanglement in the MBL phase is in a stark contrast to the ballistic entanglement spreading in ergodic systems (Kim and Huse, 2013) and in Yang-Baxter-integrable models (Calabrese and Cardy, 2009).”
• Entanglement Renyi entropies in holographic theories (2010, methodology): “(See [18] or [12] for the form of the full inverse Laplace transform.”
• The Page curve of Hawking radiation from semiclassical geometry (2019, background): “(See [34–39] for studies of local quenches in twodimensional CFTs.”
• Geometric entropy and edge modes of the electromagnetic field (2015, methodology): “To calculate entanglement entropy for more general regions by Euclidean methods one should instead use the replica trick [53].”
• Entanglement entropy of two disjoint intervals in conformal field theory: II (2010, background): “the response of the partition function to a scale transformation) under this uniformizing map, one obtains [3, 4] Tr ρA = cn −(c/6)(n−1/n), (2) from which the replica trick (1) leads to the famous result (c/3) log [6, 7] and to the full spectrum of ρA [8].”
• Quantum Entanglement Growth Under Random Unitary Dynamics (2016, methodology): “These scaling forms agree with the results for holographic CFTs [8] and with a heuristic application of the minimal cut formula to a regular tensor network [10].”
• Gapless Symmetry-Protected Topological Order (2017, methodology): “Using techniques of 1+1d CFT [130, 131], we demonstrate this edge theory has c = 1 for the gapless trivial case but c = 2 for the gapless SPT case.”
• Entanglement Entropy at Large Central Charge (2013, methodology): “In two spacetime dimensions at a critical point, when the subsystem is a single interval, the ground state entanglement entropy is universal in the sense that it does not depend on the details of the conformal field theory [1, 2, 3].”
• Evolution of holographic entanglement entropy after thermal and electromagnetic quenches (2010, methodology): “The quantum information community employ it as a tool for assessing such things as the computational complexity of a system (see [1] for a review).”
• Entanglement entropy: Holography and renormalization group (2018, methodology): “The condensed matter applications of entanglement being a probe of quantum critical phenomena and quantum quench dynamics is elaborated in the reviews (Calabrese and Cardy, 2009, 2016) by means of the CFT methods (see also (Laflorencie, 2016)).”
• Comments on holographic entanglement entropy and RG flows (2012, methodology): “Further, for a two-dimensional CFT, the entanglement entropy is given by [3, 4]”