Entanglement dualities in supersymmetry

Open Access

Entanglement dualities in supersymmetry

Robert H. Jonsson, Lucas Hackl, and Krishanu Roychowdhury

Phys. Rev. Research 3, 023213 – Published 16 June 2021

Abstract

We derive a general relation between the bosonic and fermionic entanglement in the ground states of supersymmetric quadratic Hamiltonians. For this, we construct canonical identifications between bosonic and fermionic subsystems. Our derivation relies on a unified framework to describe both bosonic and fermionic Gaussian states in terms of so-called linear complex structures $J$ . The resulting dualities apply to the full entanglement spectrum between the bosonic and the fermionic systems, such that the von Neumann entropy and arbitrary Renyi entropies can be related. We illustrate our findings in one- and two-dimensional systems, including the paradigmatic Kitaev honeycomb model. While typically supersymmetry preserves features like area law scaling of the entanglement entropies on either side, we find a peculiar phenomenon, namely, an amplified scaling of the entanglement entropy (“super area law”) in bosonic subsystems when the dual fermionic subsystems develop almost maximally entangled modes.

Received 22 March 2021
Revised 6 May 2021
Accepted 7 May 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.023213

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum correlations in quantum information Quantum entanglement

Supersymmetry

Quantum Information, Science & TechnologyGeneral PhysicsParticles & FieldsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Robert H. Jonsson ^1,*, Lucas Hackl ^2,3,†, and Krishanu Roychowdhury ^4,‡

¹Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
²School of Mathematics and Statistics & School of Physics, The University of Melbourne, Parkville, Victoria 3010, Australia
³QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
⁴Department of Physics, Stockholm University, SE-106 91 Stockholm, Sweden

^*robert.jonsson@mpq.mpg.de
^†lucas.hackl@unimelb.edu.au
^‡krishanu.1987@gmail.com

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Vol. 3, Iss. 2 — June - August 2021

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Images

Figure 1
Spectrum of the Kitaev chain with open ends. The system is in a trivial phase for $| t / μ | < 1 / 2$ , critical at $t / μ = \pm 1 / 2$ , and topological otherwise, with edge modes appearing.
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Figure 2
Locality of the identification map $L_{1}$ and its dependence on the relative coupling $t / μ$ . As schematically visualized in (b), the plots show, for a system of $N = 30$ modes, how $L_{1}$ associates the onsite operator ${\hat{c}}_{15}$ in the fermionic Kitaev chain in Eq. (53) to the operator $L_{1} ({\hat{c}}_{15}) = \sum_{j} α_{j} {\hat{b}}_{j} + β_{j} {\hat{b}}_{j}^{†}$ on the bosonic Kane-Lubensky chain in Eq. (55). In the trivial phase, as plotted on the left in (a) for $t = 0.35 μ$ , the identification map preserves locality to a very high degree, namely, with an exponential decay of the coefficients $| α_{j} |^{2}$ and $| β_{j} |^{2}$ with the distance $| k - j |$ (here $k \equiv 15$ ). In the topological phase, as plotted on the right in (c) for $t = μ$ , the operator $L_{1} ({\hat{c}}_{k})$ can be nonlocal with a strong contribution from the boundary sites.
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Figure 3
Entanglement entropies for dual subsystems each consisting of a single mode. The solid line shows the entanglement entropy of a single fermionic mode for which the restricted complex structure has eigenvalues $\pm i λ$ . Due to the entanglement duality in Eq. (80), the restricted complex structure of the dual bosonic mode has eigenvalues $\pm i λ^{- 1}$ , and the dashed line plots the resulting entanglement entropy.
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Figure 4
Supersymmetric Kitaev honeycomb model: The schematic visualization in (a) represents, on the left, the fermionic honeycomb Kitaev model in Eq. (105) and, on the right, the bosonic triangular lattice in Eq. (107), which are generated by the supercharge in Eq. (106). The map $L_{1}^{- 1}$ identifies fermionic subsystems with bosonic dual subsystems. The plots below, in (b) and (c), show numerical calculations of the entanglement entropy of parallelogram-shaped subsystems on the fermionic side and of their dual bosonic subsystems on the other side for two different orientations of the hopping parameters $\vec{j} = (j_{x}, j_{y}, j_{z})$ . For the numerical examples, periodic lattices with a total number of $N = 45 \times 45 = 2025$ unit cells were considered. The parallelograms of the subsystems contain $M = m \times m$ modes, i.e., $M$ unit cells of the honeycomb lattice, and are oriented such that they do not cut through links with hopping parameter $j_{x}$ . The fermionic entropies show good numerical agreement with the area law, which they are known to follow in the thermodynamic limit. Depending on the orientation of the couplings $\vec{j}$ relative to the parallelogram, the dual entropies can follow the area law or a scale much faster.
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Figure 5
Absolute values of the eigenvalues of the fermionic linear complex structure $J_{A}^{f}$ on the Kitaev honeycomb lattice restricted to the subsystems of Fig. 4. The inset zooms in on the eigenvalues of the first case $\vec{j} = (2.5, 1, 1)$ , which hardly fall below $| λ_{i} | \approx 0.9$ . In the second case $\vec{j} = (1, 1, 2.5)$ , the absolute values decay exponentially with the subsystem size $M$ , thus triggering the amplified scaling of the dual entropies in Fig. 4 up until the subsystem exhausts the full lattice of $N = 2025$ modes.
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Figure 6
Stacked plots of the mode-wise contribution to the total entanglement entropy, for the setup of Figs. 4 and 4, for the case $\vec{j} = (1, 1, 2.5)$ . The left plot refers to the fermionic hexagonal subsystems $A_{f}$ and the right plot to the dual bosonic subsystem $L_{1}^{- 1} (A_{f})$ . The difference between the $(i - 1) th$ and $i th$ line gives the contribution of the $i th$ normal mode of the subsystem to the total entanglement entropy. Thus, the upper-most line coincides with the respective plot in Figs. 4 and 4. Note that, here, we have changed the horizontal axis to $\sqrt{M}$ , i.e.,the side length of the parallelogram-shaped fermionic subsystems. The fermionic entropy is dominated by the $\sqrt{M}$ normal modes that are almost maximally entangled and the bosonic entropy by their dual modes.
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