arXiv:0802.3231 · Journal of High Energy Physics

Topological entanglement entropy in Chern-Simons theories and quantum Hall fluids

Shiying Dong, Eduardo Fradkin, Robert G Leigh, Sean Nowling

Published May 7, 2008• 218 citations

DOI: 10.1088/1126-6708/2008/05/016

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arXiv:0802.3231 · Journal of High Energy Physics

Topological entanglement entropy in Chern-Simons theories and quantum Hall fluids

Shiying Dong, Eduardo Fradkin, Robert G Leigh, Sean Nowling

Published May 7, 2008• 218 citations

arXiv HTML PDF

DOI: 10.1088/1126-6708/2008/05/016

Related papers

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Andrew Stewart Caldin

FINDING: Chern-Simons theory at level $k$ governs fractional quantum Hall anyons; braiding produces non-abelian phases encoded in the Jones polynomial; lattice realizations confirm topological invariance; a novel CKM angle measurement is unrelated to topological physics. MATH: Chern-Simons action $S_{CS} = \frac{k}{4\pi}\int \text{Tr}(A\wedge dA + \frac{2}{3}A\wedge A\wedge A)$ ; level $k \in \mathbb{Z}$ (integer QHE) or $k = p/q$ (fractional QHE, e.g., $k=1/3$ for Laughlin state); anyon braiding phase $\theta = \pi/k$ (abelian) or unitary matrix $U_{ab}$ (non-abelian, e.g., Fibonacci anyons with $U \sim e^{i\pi/5}$ ); entanglement entropy $S_A = \alpha L - \gamma$ , with topological entanglement entropy $\gamma = \ln \sqrt{D}$ , $D = \sum_i d_i^2$ (quantum dimensions); lattice Chern-Simons reproduces continuum in limit $a \to 0$ with $k$ quantized. CONNECTION: Level $k=5$ gives Fibonacci anyons with quantum dimension $d = \phi = 1.618$ (golden ratio) — di Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Chern-Simons theory at level $k$ governs fractional quantum Hall anyons; braiding produces non-abelian phases encoded in the Jones polynomial; lattice realizations confirm topological invariance; a novel CKM angle measurement is unrelated to topological physics. MATH: Chern-Simons action $S_{CS} = \frac{k}{4\pi}\int \text{Tr}(A\wedge dA + \frac{2}{3}A\wedge A\wedge A)$; level $k \in \mathbb{Z}$ (integer QHE) or $k = p/q$ (fractional QHE, e.g., $k=1/3$ for Laughlin state); anyon braiding phase $\theta = \pi/k$ (abelian) or unitary matrix $U_{ab}$ (non-abelian, e.g., Fibonacci anyons with $U \sim e^{i\pi/5}$); entanglement entropy $S_A = \alpha L - \gamma$, with topological entanglement entropy $\gamma = \ln \sqrt{D}$, $D = \sum_i d_i^2$ (quantum dimensions); lattice Chern-Simons reproduces continuum in limit $a \to 0$ with $k$ quantized. CONNECTION: Level $k=5$ gives Fibonacci anyons with quantum dimension $d = \phi = 1.618$ (golden ratio) — di Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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FINDING: Chern-Simons theory at level

k

governs fractional quantum Hall anyons; braiding produces non-abelian phases encoded in the Jones polynomial; lattice realizations confirm topological invariance; a novel CKM angle measurement is unrelated to topological physics. MATH: Chern-Simons action

S_{CS} = \frac{k}{4\pi}\int \text{Tr}(A\wedge dA + \frac{2}{3}A\wedge A\wedge A)

; level

k \in \mathbb{Z}

(integer QHE) or

k = p/q

(fractional QHE, e.g.,

k=1/3

for Laughlin state); anyon braiding phase

\theta = \pi/k

(abelian) or unitary matrix

U_{ab}

(non-abelian, e.g., Fibonacci anyons with

U \sim e^{i\pi/5}

); entanglement entropy

S_A = \alpha L - \gamma

, with topological entanglement entropy

\gamma = \ln \sqrt{D}

D = \sum_i d_i^2

(quantum dimensions); lattice Chern-Simons reproduces continuum in limit

a \to 0

with

k

quantized. CONNECTION: Level

k=5

gives Fibonacci anyons with quantum dimension

d = \phi = 1.618

(golden ratio) — di Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com