Entanglement entropy and conformal field theory

Calabrese, Pasquale; Cardy, John

doi:10.1088/1751-8113/42/50/504005

Cited by 1,591 publications

(2,342 citation statements)

References 190 publications

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“…V). "Thermalization is slower than operator spreading" in generic 1D systems (i.e., in general ,v E is smaller than v B ): by contrast, this is not true in 1+1D CFTs [2], or in certain toy models [23]. It would be interesting to make more detailed comparisons with holographic models [8].…”

Section: Discussionmentioning

confidence: 99%

“…To generalize the conjecture to systems without noise, we must allow for the fact that the asymptotic value of the entanglement depends on the energy density of the initial state. We therefore replace the entanglement S in the formulas with S=s eq , where s eq is the equilibrium entropy density corresponding to the initial energy density [2,8]. This ensures that the entanglement entropy of an l-sized region matches the equilibrium thermal entropy when v E t ≫ l=2, as required for thermalization.…”

Section: A Scaling Forms For the Entanglement Tsunamimentioning

confidence: 99%

“…This irreversible growth of entanglementquantified by the growth of the von Neumman entropyis important for several reasons. It is an essential part of thermalization, and as a result has been addressed in diverse contexts ranging from conformal field theory [1][2][3][4] and holography [5][6][7][8][9][10][11][12] to integrable [13][14][15][16][17][18][19], nonintegrable [20][21][22][23], and strongly disordered spin chains [24][25][26][27][28][29][30]. Entanglement growth is also of practical importance as the crucial obstacle to simulating quantum dynamics numerically, for example, using matrix product states or the density matrix renormalization group [31].…”

Section: Introductionmentioning

confidence: 99%

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Quantum Entanglement Growth under Random Unitary Dynamics

et al. 2017

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Characterizing how entanglement grows with time in a many-body system, for example, after a quantum quench, is a key problem in nonequilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time-dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the "entanglement tsunami" in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow according to the celebrated KardarParisi-Zhang (KPZ) equation. The mean entanglement grows linearly in time, while fluctuations grow like ðtimeÞ 1=3 and are spatially correlated over a distance ∝ ðtimeÞ 2=3 . We derive KPZ universal behavior in three complementary ways, by mapping random entanglement growth to (i) a stochastic model of a growing surface, (ii) a "minimal cut" picture, reminiscent of the Ryu-Takayanagi formula in holography, and (iii) a hydrodynamic problem involving the dynamical spreading of operators. We demonstrate KPZ universality in 1D numerically using simulations of random unitary circuits. Importantly, the leading-order time dependence of the entropy is deterministic even in the presence of noise, allowing us to propose a simple coarse grained minimal cut picture for the entanglement growth of generic Hamiltonians, even without noise, in arbitrary dimensionality. We clarify the meaning of the "velocity" of entanglement growth in the 1D entanglement tsunami. We show that in higher dimensions, noisy entanglement evolution maps to the wellstudied problem of pinning of a membrane or domain wall by disorder.

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Section: Discussionmentioning

confidence: 99%

Section: A Scaling Forms For the Entanglement Tsunamimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quantum Entanglement Growth under Random Unitary Dynamics

et al. 2017

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“…From the path integral point of view, the calculation of Trρ N A is equivalent to find the partition function on a N -sheet Riemann surface which glued together along B but leave A cut open [1] Trρ 3) where Z N (A) is the partition function on the N -sheeted Riemann surface and the normalization factor Z is just the original partition function…”

Section: Jhep01(2016)058mentioning

confidence: 99%

“…It is defined as the von Neumann entropy of the reduced density matrix of the subsystem. For dimension D = 2, one can use the tools of CFT, see [1] for a review. Essentially, to obtain the entanglement entropy is equivalent to calculate the partition function of certain CFTs on the higher genus Riemann surface.…”

Section: Introductionmentioning

confidence: 99%

Two intervals Rényi entanglement entropy of compact free boson on torus

Liu

2016

J. High Energ. Phys.

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Abstract:We compute the N = 2 Rényi entanglement entropy of two intervals at equal time in a circle, for the theory of a 2D compact complex free scalar at finite temperature. This is carried out by performing functional integral on a genus 3 ramified cover of the torus, wherein the quantum part of the integral is captured by the four point function of twist fields on the worldsheet torus, and the classical piece is given by summing over winding modes of the genus 3 surface onto the target space torus. The final result is given in terms of a product of theta functions and certain multi-dimensional theta functions. We demonstrate the T-duality invariance of the result. We also study its low temperature limit. In the case in which the size of the intervals and of their separation are much smaller than the whole system, our result is in exact agreement with the known result for two intervals on an infinite system at zero temperature [5]. In the case in which the separation between the two intervals is much smaller than the interval length, the leading thermal corrections take the same universal form as proposed in [9,10] for Rényi entanglement entropy of a single interval.

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Entanglement in fermionic chains with interface defects

Eisler

Peschel

2010

Annalen der Physik

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We study the ground-state entanglement of two halves of a critical transverse Ising chain, separated by an interface defect. From the relation to a two-dimensional Ising model with a defect line we obtain an exact expression for the continuously varying effective central charge which governs the asymptotic behaviour of the entanglement entropy. The result is relevant also for other fermionic chains.

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Entanglement entropy and conformal field theory

Cited by 1,591 publications

References 190 publications

Quantum Entanglement Growth under Random Unitary Dynamics

Quantum Entanglement Growth under Random Unitary Dynamics

Two intervals Rényi entanglement entropy of compact free boson on torus

Entanglement in fermionic chains with interface defects

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