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Superconformal Models for Graphene and Boundary Central Charges

Christopher P. Herzog, Kuo-Wei Huang, Itamar Shamir, Julio Virrueta

TL;DR

This work demonstrates that graphene-like boundary conformal field theories can be extended to supersymmetric settings with ${ m math{N}=1,2,4}$ in the bulk. The bulk gauge coupling remains exactly marginal to all orders, while boundary central charges $b_1$ and $b_2$ depend perturbatively on the coupling, contrasting with the bulk $c$-charge which is coupling-independent under SUSY. A nonzero $ heta$-term is shown to renormalize boundary conditions via a cos α–scaling of the effective coupling. In the special ${ m N}=4$ case, the combination $b_1-b_2$ may be coupling-independent at one loop, suggesting a potential deeper structure in the boundary displacement multiplet and associated curvature invariants.

Abstract

In the context of boundary conformal field theory, we investigate whether the boundary trace anomaly can depend on marginal directions in the presence of supersymmetry. Recently, it was found that a graphene-like non-supersymmetric conformal field theory with a four-dimensional bulk photon and a three-dimensional boundary electron has two boundary central charges that depend on an exactly marginal direction, namely the gauge coupling. In this work, we supersymmetrize this theory, paying special attention to the boundary terms required by supersymmetry. We study models with 4, 8, and 16 Poincaré supercharges in the bulk, half of which are broken by the boundary. In all cases, we find that at all orders in perturbation theory, the gauge coupling is not renormalized, providing strong evidence that these theories are boundary conformal field theories. Moreover, the boundary central charges depend on the coupling. One possible exception to this dependence on marginal directions is that the difference between the two charges is coupling independent at one-loop in the maximally supersymmetric case. In our analysis, a possible boundary Chern-Simons term is incorporated by a bulk $θ$-term.

Superconformal Models for Graphene and Boundary Central Charges

TL;DR

This work demonstrates that graphene-like boundary conformal field theories can be extended to supersymmetric settings with in the bulk. The bulk gauge coupling remains exactly marginal to all orders, while boundary central charges and depend perturbatively on the coupling, contrasting with the bulk -charge which is coupling-independent under SUSY. A nonzero -term is shown to renormalize boundary conditions via a cos α–scaling of the effective coupling. In the special case, the combination may be coupling-independent at one loop, suggesting a potential deeper structure in the boundary displacement multiplet and associated curvature invariants.

Abstract

In the context of boundary conformal field theory, we investigate whether the boundary trace anomaly can depend on marginal directions in the presence of supersymmetry. Recently, it was found that a graphene-like non-supersymmetric conformal field theory with a four-dimensional bulk photon and a three-dimensional boundary electron has two boundary central charges that depend on an exactly marginal direction, namely the gauge coupling. In this work, we supersymmetrize this theory, paying special attention to the boundary terms required by supersymmetry. We study models with 4, 8, and 16 Poincaré supercharges in the bulk, half of which are broken by the boundary. In all cases, we find that at all orders in perturbation theory, the gauge coupling is not renormalized, providing strong evidence that these theories are boundary conformal field theories. Moreover, the boundary central charges depend on the coupling. One possible exception to this dependence on marginal directions is that the difference between the two charges is coupling independent at one-loop in the maximally supersymmetric case. In our analysis, a possible boundary Chern-Simons term is incorporated by a bulk -term.

Paper Structure

This paper contains 17 sections, 123 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Ultrarelativistic Models of Graphene
  3. Non-Supersymmetric Model
  4. ${\mathcal{N}}=1$ Super Graphene
  5. ${\mathcal{N}}=2$ Super Graphene
  6. Propagators and Displacement Operator Correlators
  7. Scalar
  8. Photon
  9. Photino
  10. Relations between $\Omega_{\rm s}$, $\Omega_{\rm v}$ and $\Omega_{\rm f}$
  11. Displacement Operator Two- and Three-Point Functions
  12. Perturbation Theory for Super Graphene
  13. Renormalization Group Analysis
  14. One Loop Calculations
  15. Conclusion and Future Perspective
  16. ...and 2 more sections

Figures (9)

  • Figure 1: The photon self-energy at one-loop.
  • Figure 2: The photino self-energy at one-loop.
  • Figure 3: (a) contributes to the $X$ self-energy; (b), (c) contribute to the $Y$ self-energy.
  • Figure 4: The electron self-energy at one-loop in the ${\mathcal{N}}=1$ theory.
  • Figure 5: The selectron self-energy at one-loop in the ${\mathcal{N}}=1$ theory.
  • ...and 4 more figures