A Duality Web in 2+1 Dimensions and Condensed Matter Physics (1606.01989v2)

Abstract: Building on earlier work in the high energy and condensed matter communities, we present a web of dualities in $2+1$ dimensions that generalize the known particle/vortex duality. Some of the dualities relate theories of fermions to theories of bosons. Others relate different theories of fermions. For example, the long distance behavior of the $2+1$-dimensional analog of QED with a single Dirac fermion (a theory known as $U(1)_{1/2}$) is identified with the $O(2)$ Wilson-Fisher fixed point. The gauged version of that fixed point with a Chern-Simons coupling at level one is identified as a free Dirac fermion. The latter theory also has a dual version as a fermion interacting with some gauge fields. Assuming some of these dualities, other dualities can be derived. Our analysis resolves a number of confusing issues in the literature including how time reversal is realized in these theories. It also has many applications in condensed matter physics like the theory of topological insulators (and their gapped boundary states) and the problem of electrons in the lowest Landau level at half filling. (Our techniques also clarify some points in the fractional Hall effect and its description using flux attachment.) In addition to presenting several consistency checks, we also present plausible (but not rigorous) derivations of the dualities and relate them to $3+1$-dimensional $S$-duality.

• The paper introduces a duality framework linking bosonic and fermionic theories in 2+1 dimensions to explain condensed matter phenomena.
• It employs theoretical consistency checks and derivations to extend particle/vortex duality concepts to systems like topological insulators.
• The study’s findings deepen understanding of quantum Hall effects and half-filled Landau levels, opening avenues for further research.

An Expert Review of "A Duality Web in 2 + 1 Dimensions and Condensed Matter Physics"

The paper "A Duality Web in 2 + 1 Dimensions and Condensed Matter Physics," authored by Nathan Seiberg, T. Senthil, Chong Wang, and Edward Witten, explores a sophisticated web of dualities in 2+1 dimensions, extending established concepts like particle/vortex duality. This research establishes new relationships between theories involving both bosons and fermions, and offers insights into condensed matter phenomena such as topological insulators and the half-filled Landau level.

Core Contributions

The paper primarily introduces a comprehensive framework of dualities that spans both high-energy physics and condensed matter. It systematically derives several duality correspondences by starting with known relationships, notably highlighting the connection between the long-range behavior of 2+1-dimensional QED with a single Dirac fermion and the O(2) Wilson-Fisher fixed point. Importantly, the authors demonstrate a bold claim: the gauged version of this fixed point, with a Chern-Simons coupling at level one, corresponds to a free Dirac fermion. This demonstrates that the intricate web of dualities is not limited to supersymmetric cases but extends to non-supersymmetric theories in 2+1 dimensions.

Methodology and Results

The authors employ both theoretical consistency checks and plausible (though non-rigorous) derivations to fortify their duality claims. The paper navigates through the complexities of high-energy terminology to elucidate the duality relationships clearly, showing how distinct presentations of theories reveal identical physical behavior. Particularly noteworthy is their analysis resolving ambiguous issues in the literature regarding time reversal symmetry and its realization in these systems.

An essential component of the work is the linkage to condensed matter applications. By leveraging the generalized dualities, the authors provide valuable insights into phenomena like the fractional quantum Hall effect and electron behavior in topological insulators. They interpret the complex behavior of electrons in the lowest Landau level at half filling through these dualities, offering potential enhancements to previous theoretical frameworks, such as the HLR theory.

Implications and Future Directions

The implications of this paper are significant, both theoretically and practically. The paper proposes that an understanding of these dualities can lead to novel insights into otherwise challenging problems in condensed matter systems, potentially guiding future experimental investigations. It also opens pathways to extending these duality principles to 3+1-dimensional theories, as evidenced by the exploration of S-duality relationships.

Future research can benefit from this foundational work by exploring duality transformations in higher-dimensional field theories or leveraging these results to explore the properties of quantum Hall systems and other strongly correlated electron systems. The authors speculate that further extending these dualities might resolve outstanding questions in condensed matter physics, offering new strategies for tackling complex quantum phenomena.

Conclusion

In summary, the paper by Seiberg et al. offers a vital contribution to understanding dualities in lower-dimensional field theories and their connections to tangible physical systems. By extending known dualities and deriving new ones, the research presented provides an enriched framework for interpreting a variety of quantum phenomena. The potential for future advances grounded in these insights is substantial, providing an anchor for ongoing studies in both theoretical and experimental physics.