Toward a New Theory of the Fractional Quantum Hall Effect
The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>σ</mi><mrow><mi...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-01-01
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| Series: | Nanomaterials |
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| Online Access: | https://www.mdpi.com/2079-4991/14/3/297 |
| _version_ | 1797318369432567808 |
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| author | Sergey A. Mikhailov |
| author_facet | Sergey A. Mikhailov |
| author_sort | Sergey A. Mikhailov |
| collection | DOAJ |
| description | The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub></semantics></math></inline-formula> of a two-dimensional electron system is quantized, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msup><mi>e</mi><mn>2</mn></msup><mo>/</mo><mn>3</mn><mi>h</mi></mrow></semantics></math></inline-formula>, in the vicinity of the Landau level filling factor <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>3</mn></mrow></semantics></math></inline-formula>. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of <i>N</i> two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>≤</mo><mn>7</mn></mrow></semantics></math></inline-formula> electrons continuously as a function of magnetic field in the range <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mn>4</mn><mo>≲</mo><mi>ν</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the <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>3</mn></mrow></semantics></math></inline-formula> Laughlin state for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>≤</mo><mn>8</mn></mrow></semantics></math></inline-formula> and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small <i>N</i> nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect. |
| first_indexed | 2024-03-08T03:51:27Z |
| format | Article |
| id | doaj.art-209d381c06414b9ca13e4e22cfc07cde |
| institution | Directory Open Access Journal |
| issn | 2079-4991 |
| language | English |
| last_indexed | 2024-03-08T03:51:27Z |
| publishDate | 2024-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Nanomaterials |
| spelling | doaj.art-209d381c06414b9ca13e4e22cfc07cde2024-02-09T15:19:30ZengMDPI AGNanomaterials2079-49912024-01-0114329710.3390/nano14030297Toward a New Theory of the Fractional Quantum Hall EffectSergey A. Mikhailov0Institute of Physics, University of Augsburg, D-86135 Augsburg, GermanyThe fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub></semantics></math></inline-formula> of a two-dimensional electron system is quantized, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>σ</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><msup><mi>e</mi><mn>2</mn></msup><mo>/</mo><mn>3</mn><mi>h</mi></mrow></semantics></math></inline-formula>, in the vicinity of the Landau level filling factor <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>3</mn></mrow></semantics></math></inline-formula>. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of <i>N</i> two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>≤</mo><mn>7</mn></mrow></semantics></math></inline-formula> electrons continuously as a function of magnetic field in the range <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mn>4</mn><mo>≲</mo><mi>ν</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the <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>3</mn></mrow></semantics></math></inline-formula> Laughlin state for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>≤</mo><mn>8</mn></mrow></semantics></math></inline-formula> and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small <i>N</i> nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect.https://www.mdpi.com/2079-4991/14/3/297fractional quantum Hall effectexact diagonalizationtwo-dimensional electron gaselectron–electron interaction |
| spellingShingle | Sergey A. Mikhailov Toward a New Theory of the Fractional Quantum Hall Effect Nanomaterials fractional quantum Hall effect exact diagonalization two-dimensional electron gas electron–electron interaction |
| title | Toward a New Theory of the Fractional Quantum Hall Effect |
| title_full | Toward a New Theory of the Fractional Quantum Hall Effect |
| title_fullStr | Toward a New Theory of the Fractional Quantum Hall Effect |
| title_full_unstemmed | Toward a New Theory of the Fractional Quantum Hall Effect |
| title_short | Toward a New Theory of the Fractional Quantum Hall Effect |
| title_sort | toward a new theory of the fractional quantum hall effect |
| topic | fractional quantum Hall effect exact diagonalization two-dimensional electron gas electron–electron interaction |
| url | https://www.mdpi.com/2079-4991/14/3/297 |
| work_keys_str_mv | AT sergeyamikhailov towardanewtheoryofthefractionalquantumhalleffect |