Effect of single impurity on free fermion entanglement entropy

Mohammad Pouranvari, Kun Yang, and Alexander Seidel

Phys. Rev. B 91, 075115 – Published 18 February 2015

Abstract

The one-dimensional free Fermi gas is a prototype conformally invariant system whose entanglement properties are well understood. In this paper, the effects of a single impurity on one-dimensional free fermion entanglement entropy are studied both analytically and numerically. Such an impurity represents an exactly marginal perturbation to the bulk conformally invariant fixed point. We find that the impurity leads to subleading contributions to the entanglement entropy that scale inversely with the subsystem size. The origins of such contributions are identified.

Received 19 December 2014
Revised 27 January 2015

DOI:https://doi.org/10.1103/PhysRevB.91.075115

Authors & Affiliations

Mohammad Pouranvari and Kun Yang

National High Magnetic Field Laboratory and Department of Physics, Florida State University, Tallahassee, Florida 32306, USA

Alexander Seidel

Department of Physics, Washington University, St. Louis, Missouri 63160, USA

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Vol. 91, Iss. 7 — 15 February 2015

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Figure 1
Panel (a): Free fermion entanglement entropy in log-linear scale. We set $N_{A} = N_{F} = N / 10 + 1$ , and $N$ goes from 100 to 17000 in increments of 100. Blue dots are numerical results and the red line is a fitted line with a slope of $0.3315$ and a root mean square error (RMSE) of 0.0014. The slope is in agreement with an analytical value of $1 / 3$ . Panel (b): main plot is $EE - 1 / 3 log N_{A}$ in linear-linear scale with the same setting of panel (a). $EE - 1 / 3 log N_{A}$ approaches $0.3293$ for big subsystem size, close to the analytic value of $0.3401$ . The inset plot is $EE - 1 / 3 ln N_{A} - 0.3293$ versus $N_{A}$ in a log-log scale. Slope of the fitted line is $- 2.0270$ and with a RMSE of $0.0014$ .
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Figure 2
Main plot: entanglement entropy corresponding to different values of impurity strength $w$ . Free fermion case is plotted in red. Small range of subsystem size is chosen to see the deviations of single impurity entanglement entropies. The inset is a log-linear plot of entanglement entropy for different values of impurity strength including $w = 0$ . Slope of fitted line is the same for all values of $w$ and it is $0.3315$ . In both plots, we set $N_{F} = N_{A} = N / 10 + 1$ . $N$ goes from 100 to 17000 in increments of 100.
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Figure 3
The absolute value of the difference between single impurity and free fermion entanglement entropy $| Δ EE |$ for different values of $w$ in log-log scale. Besides oscillation, absolute difference goes to zero with slope approximately equal to $- 1$ . The slope and root mean square error (RMSE) are indicated for each plot. We set $N_{F} = N_{A} = N / 10 + 1$ . $N$ goes from 100 to 17000 in increments of 100.
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Figure 4
Subsystem size $N_{A}$ times difference between single impurity and free fermion entanglement entropy, $N_{A} \times ({EE}_{w} - {EE}_{0})$ when $N_{A}$ is changing. We set $N = 4000, N_{F} = 41$ in panel (a), $N = 4000, N_{F} = 81$ in panel (b), and $N = 2000, N_{F} = 21$ in panel (c). Panel (d): $2 π / k_{F}$ versus calculated oscillation periodicity $T$ for $N_{A} \times ({EE}_{w} - {EE}_{0})$ when $N_{A}$ is changing. We set $N = 5000, N_{F} = 51, 101, 201, 501, 1001$ , for different values of impurity strength $w$ . Slope of the fitted line for different $w$ 's is $\approx 1$ .
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Figure 5
Panel (a): the oscillation amplitude $α$ (blue) and nonoscillating term $β$ (red) of $N_{A} \times ({EE}_{w} - {EE}_{0})$ for a constant $k_{F} = π / 100$ corresponding to $w = 10^{2}, 10^{3}, 10^{4}$ . Panel (b): The nonoscillating term in a log-log scale. Slope of the three $w$ 's is approximately the same: $\approx - 0.87$ . Panel (c): the oscillation amplitude $α$ of $N_{A} \times ({EE}_{w} - {EE}_{0})$ for a constant $k_{F} = π / 100$ corresponding to $w = 10^{- 2}, 10^{- 3}, 10^{- 4}$ from top to bottom. Inset plot is the saturated value of $α$ versus $w$ . Panel (d): absolute value of the nonoscillating term $| β |$ of $N_{A} \times ({EE}_{w} - {EE}_{0})$ for a constant $k_{F} = π / 100$ corresponding to $w = 10^{- 2}, 10^{- 3}, 10^{- 4}$ from top to bottom. Inset plot is the saturated value of $β$ versus $w$ .
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Figure 6
Panel (a): the oscillation amplitude $α$ (blue) and nonoscillating term $β$ (red) of $N_{A} \times ({EE}_{w} - {EE}_{0})$ for a constant $k_{F} = π / 100$ corresponding to $w = 0.1, 0.2, 0.3, 0.4, 0.5$ , from bottom to top for $α$ and from top to bottom for $β$ . Panel (b): Comparison between the saturated valued of $β$ and Eq. (28).
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