Tensor Network Renormalization with Fusion Charges—Applications to 3D Lattice Gauge Theory
Tensor network methods are powerful and efficient tools for studying the properties and dynamics of statistical and quantum systems, in particular in one and two dimensions. In recent years, these methods have been applied to lattice gauge theories, yet these theories remain a challenge in <inlin...
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MDPI AG
2020-07-01
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| Online Access: | https://www.mdpi.com/2218-1997/6/7/97 |
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| author | William J. Cunningham Bianca Dittrich Sebastian Steinhaus |
| author_facet | William J. Cunningham Bianca Dittrich Sebastian Steinhaus |
| author_sort | William J. Cunningham |
| collection | DOAJ |
| description | Tensor network methods are powerful and efficient tools for studying the properties and dynamics of statistical and quantum systems, in particular in one and two dimensions. In recent years, these methods have been applied to lattice gauge theories, yet these theories remain a challenge in <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> dimensions. In this article, we present a new (decorated) tensor network algorithm, in which the tensors encode the lattice gauge amplitude expressed in the fusion basis. This has several advantages—firstly, the fusion basis does diagonalize operators measuring the magnetic fluxes and electric charges associated to a hierarchical set of regions. The algorithm allows therefore a direct access to these observables. Secondly the fusion basis is, as opposed to the previously employed spin network basis, stable under coarse-graining. Thirdly, due to the hierarchical structure of the fusion basis, the algorithm does implement predefined disentanglers. We apply this new algorithm to lattice gauge theories defined for the quantum group <inline-formula> <math display="inline"> <semantics> <mrow> <mi>SU</mi> <msub> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mi mathvariant="normal">k</mi> </msub> </mrow> </semantics> </math> </inline-formula> and identify a weak and a strong coupling phase for various levels <inline-formula> <math display="inline"> <semantics> <mi mathvariant="normal">k</mi> </semantics> </math> </inline-formula>. As we increase the level <inline-formula> <math display="inline"> <semantics> <mi mathvariant="normal">k</mi> </semantics> </math> </inline-formula>, the critical coupling <inline-formula> <math display="inline"> <semantics> <msub> <mi>g</mi> <mi>c</mi> </msub> </semantics> </math> </inline-formula> decreases linearly, suggesting the absence of a deconfining phase for the continuous group <inline-formula> <math display="inline"> <semantics> <mrow> <mi>SU</mi> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. Moreover, we illustrate the scaling behaviour of the Wilson loops in the two phases. |
| first_indexed | 2024-03-10T18:31:29Z |
| format | Article |
| id | doaj.art-584ff20944064f60a5ec8b20a0765f39 |
| institution | Directory Open Access Journal |
| issn | 2218-1997 |
| language | English |
| last_indexed | 2024-03-10T18:31:29Z |
| publishDate | 2020-07-01 |
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| spelling | doaj.art-584ff20944064f60a5ec8b20a0765f392023-11-20T06:33:10ZengMDPI AGUniverse2218-19972020-07-01679710.3390/universe6070097Tensor Network Renormalization with Fusion Charges—Applications to 3D Lattice Gauge TheoryWilliam J. Cunningham0Bianca Dittrich1Sebastian Steinhaus2Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, CanadaPerimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, CanadaTheoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, GermanyTensor network methods are powerful and efficient tools for studying the properties and dynamics of statistical and quantum systems, in particular in one and two dimensions. In recent years, these methods have been applied to lattice gauge theories, yet these theories remain a challenge in <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> dimensions. In this article, we present a new (decorated) tensor network algorithm, in which the tensors encode the lattice gauge amplitude expressed in the fusion basis. This has several advantages—firstly, the fusion basis does diagonalize operators measuring the magnetic fluxes and electric charges associated to a hierarchical set of regions. The algorithm allows therefore a direct access to these observables. Secondly the fusion basis is, as opposed to the previously employed spin network basis, stable under coarse-graining. Thirdly, due to the hierarchical structure of the fusion basis, the algorithm does implement predefined disentanglers. We apply this new algorithm to lattice gauge theories defined for the quantum group <inline-formula> <math display="inline"> <semantics> <mrow> <mi>SU</mi> <msub> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mi mathvariant="normal">k</mi> </msub> </mrow> </semantics> </math> </inline-formula> and identify a weak and a strong coupling phase for various levels <inline-formula> <math display="inline"> <semantics> <mi mathvariant="normal">k</mi> </semantics> </math> </inline-formula>. As we increase the level <inline-formula> <math display="inline"> <semantics> <mi mathvariant="normal">k</mi> </semantics> </math> </inline-formula>, the critical coupling <inline-formula> <math display="inline"> <semantics> <msub> <mi>g</mi> <mi>c</mi> </msub> </semantics> </math> </inline-formula> decreases linearly, suggesting the absence of a deconfining phase for the continuous group <inline-formula> <math display="inline"> <semantics> <mrow> <mi>SU</mi> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. Moreover, we illustrate the scaling behaviour of the Wilson loops in the two phases.https://www.mdpi.com/2218-1997/6/7/97tensor network renormalizationlattice gauge theoryquantum groupsquantum gravity |
| spellingShingle | William J. Cunningham Bianca Dittrich Sebastian Steinhaus Tensor Network Renormalization with Fusion Charges—Applications to 3D Lattice Gauge Theory Universe tensor network renormalization lattice gauge theory quantum groups quantum gravity |
| title | Tensor Network Renormalization with Fusion Charges—Applications to 3D Lattice Gauge Theory |
| title_full | Tensor Network Renormalization with Fusion Charges—Applications to 3D Lattice Gauge Theory |
| title_fullStr | Tensor Network Renormalization with Fusion Charges—Applications to 3D Lattice Gauge Theory |
| title_full_unstemmed | Tensor Network Renormalization with Fusion Charges—Applications to 3D Lattice Gauge Theory |
| title_short | Tensor Network Renormalization with Fusion Charges—Applications to 3D Lattice Gauge Theory |
| title_sort | tensor network renormalization with fusion charges applications to 3d lattice gauge theory |
| topic | tensor network renormalization lattice gauge theory quantum groups quantum gravity |
| url | https://www.mdpi.com/2218-1997/6/7/97 |
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