Topological BF Description of 2D Accelerated Chiral Edge Modes
In this paper, we consider the topological abelian BF theory with radial boundary on a generic 3D manifold, as we were motivated by the recently discovered <i>accelerated</i> edge modes on certain Hall systems. Our aim was to research if, where, and how the boundary keeps the memory of t...
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MDPI AG
2022-03-01
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| Online Access: | https://www.mdpi.com/2073-8994/14/4/675 |
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| author | Erica Bertolini Filippo Fecit Nicola Maggiore |
| author_facet | Erica Bertolini Filippo Fecit Nicola Maggiore |
| author_sort | Erica Bertolini |
| collection | DOAJ |
| description | In this paper, we consider the topological abelian BF theory with radial boundary on a generic 3D manifold, as we were motivated by the recently discovered <i>accelerated</i> edge modes on certain Hall systems. Our aim was to research if, where, and how the boundary keeps the memory of the details of the background metrics. We discovered that some features were topologically protected and did not depend on the bulk metric. The outcome was that these edge excitations were <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>c</mi><mi>c</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi></mrow></semantics></math></inline-formula>, as a direct consequence of the non-flat nature of the bulk spacetime. We found three possibilities for the motion of the edge quasiparticles: same directions, opposite directions, and a single-moving mode. However, requiring that the Hamiltonian of the 2D theory is bounded by below, <i>the case of the edge modes moving in the same direction was ruled out</i>. Systems involving parallel Hall currents (for instance, a fractional quantum Hall effect with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ν</mi><mo>=</mo><mn>2</mn><mo>/</mo><mn>5</mn></mrow></semantics></math></inline-formula>) cannot be described by a BF theory with the boundary, independently from the geometry of the bulk spacetime, because of positive energy considerations. Thus, we were left with physical situations characterized by edge excitations moving with opposite velocities (for example, the fractional quantum Hall effect with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ν</mi><mo>=</mo><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>n</mi></mrow></semantics></math></inline-formula>, with the <i>n</i> positive integer, and the helical Luttinger liquids phenomena) or a single-moving mode (quantum anomalous Hall). A strong restriction was obtained by requiring time reversal symmetry, which uniquely identifies modes with equal and opposite velocities, and we know that this is the case of topological insulators. The novelty, with respect to the flat bulk background, is that the modes have local velocities, which correspond to topological insulators with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>c</mi><mi>c</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi></mrow></semantics></math></inline-formula> edge modes. |
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| spelling | doaj.art-65951a11d0be4c379046a2f5acb8de0b2023-11-30T21:58:48ZengMDPI AGSymmetry2073-89942022-03-0114467510.3390/sym14040675Topological BF Description of 2D Accelerated Chiral Edge ModesErica Bertolini0Filippo Fecit1Nicola Maggiore2Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, I-16146 Genova, ItalyDipartimento di Fisica, Università di Genova, Via Dodecaneso 33, I-16146 Genova, ItalyDipartimento di Fisica, Università di Genova, Via Dodecaneso 33, I-16146 Genova, ItalyIn this paper, we consider the topological abelian BF theory with radial boundary on a generic 3D manifold, as we were motivated by the recently discovered <i>accelerated</i> edge modes on certain Hall systems. Our aim was to research if, where, and how the boundary keeps the memory of the details of the background metrics. We discovered that some features were topologically protected and did not depend on the bulk metric. The outcome was that these edge excitations were <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>c</mi><mi>c</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi></mrow></semantics></math></inline-formula>, as a direct consequence of the non-flat nature of the bulk spacetime. We found three possibilities for the motion of the edge quasiparticles: same directions, opposite directions, and a single-moving mode. However, requiring that the Hamiltonian of the 2D theory is bounded by below, <i>the case of the edge modes moving in the same direction was ruled out</i>. Systems involving parallel Hall currents (for instance, a fractional quantum Hall effect with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ν</mi><mo>=</mo><mn>2</mn><mo>/</mo><mn>5</mn></mrow></semantics></math></inline-formula>) cannot be described by a BF theory with the boundary, independently from the geometry of the bulk spacetime, because of positive energy considerations. Thus, we were left with physical situations characterized by edge excitations moving with opposite velocities (for example, the fractional quantum Hall effect with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ν</mi><mo>=</mo><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>n</mi></mrow></semantics></math></inline-formula>, with the <i>n</i> positive integer, and the helical Luttinger liquids phenomena) or a single-moving mode (quantum anomalous Hall). A strong restriction was obtained by requiring time reversal symmetry, which uniquely identifies modes with equal and opposite velocities, and we know that this is the case of topological insulators. The novelty, with respect to the flat bulk background, is that the modes have local velocities, which correspond to topological insulators with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>c</mi><mi>c</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi></mrow></semantics></math></inline-formula> edge modes.https://www.mdpi.com/2073-8994/14/4/675quantum field theoryfield theory with boundarytopological states of matter |
| spellingShingle | Erica Bertolini Filippo Fecit Nicola Maggiore Topological BF Description of 2D Accelerated Chiral Edge Modes Symmetry quantum field theory field theory with boundary topological states of matter |
| title | Topological BF Description of 2D Accelerated Chiral Edge Modes |
| title_full | Topological BF Description of 2D Accelerated Chiral Edge Modes |
| title_fullStr | Topological BF Description of 2D Accelerated Chiral Edge Modes |
| title_full_unstemmed | Topological BF Description of 2D Accelerated Chiral Edge Modes |
| title_short | Topological BF Description of 2D Accelerated Chiral Edge Modes |
| title_sort | topological bf description of 2d accelerated chiral edge modes |
| topic | quantum field theory field theory with boundary topological states of matter |
| url | https://www.mdpi.com/2073-8994/14/4/675 |
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