Tensor product representation of a topological ordered phase: Necessary symmetry conditions
The tensor product representation of quantum states leads to a promising variational approach to study quantum phase and quantum phase transitions, especially topological ordered phases which are impossible to handle with conventional methods due to their long-range entanglement. However, an importa...
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| Format: | Article |
| Language: | en_US |
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American Physical Society
2010
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| Online Access: | http://hdl.handle.net/1721.1/60355 https://orcid.org/0000-0001-7296-523X https://orcid.org/0000-0002-5874-581X |
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| author | Chuang, Isaac L. Chen, Xie Wen, Xiao-Gang Zeng, Bei Gu, Zheng-Cheng |
| author2 | Massachusetts Institute of Technology. Department of Physics |
| author_facet | Massachusetts Institute of Technology. Department of Physics Chuang, Isaac L. Chen, Xie Wen, Xiao-Gang Zeng, Bei Gu, Zheng-Cheng |
| author_sort | Chuang, Isaac L. |
| collection | MIT |
| description | The tensor product representation of quantum states leads to a promising variational approach to study quantum phase and quantum phase transitions, especially topological ordered phases which are impossible to handle with conventional methods due to their long-range entanglement. However, an important issue arises when we use tensor product states (TPSs) as variational states to find the ground state of a Hamiltonian: can arbitrary variations in the tensors that represent ground state of a Hamiltonian be induced by local perturbations to the Hamiltonian? Starting from a tensor product state which is the exact ground state of a Hamiltonian with Z[subscript 2] topological order, we show that, surprisingly, not all variations in the tensors correspond to the variation in the ground state caused by local perturbations of the Hamiltonian. Even in the absence of any symmetry requirement of the perturbed Hamiltonian, one necessary condition for the variations in the tensors to be physical is that they respect certain Z[subscript 2] symmetry. We support this claim by calculating explicitly the change in topological entanglement entropy with different variations in the tensors. This finding will provide important guidance to numerical variational study of topological phase and phase transitions. It is also a crucial step in using TPS to study universal properties of a quantum phase and its topological order. |
| first_indexed | 2024-09-23T15:39:39Z |
| format | Article |
| id | mit-1721.1/60355 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T15:39:39Z |
| publishDate | 2010 |
| publisher | American Physical Society |
| record_format | dspace |
| spelling | mit-1721.1/603552022-10-02T03:11:55Z Tensor product representation of a topological ordered phase: Necessary symmetry conditions Chuang, Isaac L. Chen, Xie Wen, Xiao-Gang Zeng, Bei Gu, Zheng-Cheng Massachusetts Institute of Technology. Department of Physics Chuang, Isaac L. Chuang, Isaac L. Chen, Xie Wen, Xiao-Gang The tensor product representation of quantum states leads to a promising variational approach to study quantum phase and quantum phase transitions, especially topological ordered phases which are impossible to handle with conventional methods due to their long-range entanglement. However, an important issue arises when we use tensor product states (TPSs) as variational states to find the ground state of a Hamiltonian: can arbitrary variations in the tensors that represent ground state of a Hamiltonian be induced by local perturbations to the Hamiltonian? Starting from a tensor product state which is the exact ground state of a Hamiltonian with Z[subscript 2] topological order, we show that, surprisingly, not all variations in the tensors correspond to the variation in the ground state caused by local perturbations of the Hamiltonian. Even in the absence of any symmetry requirement of the perturbed Hamiltonian, one necessary condition for the variations in the tensors to be physical is that they respect certain Z[subscript 2] symmetry. We support this claim by calculating explicitly the change in topological entanglement entropy with different variations in the tensors. This finding will provide important guidance to numerical variational study of topological phase and phase transitions. It is also a crucial step in using TPS to study universal properties of a quantum phase and its topological order. National Science Foundation (U.S.) (Grant No. NSFPHY05- 51164) National Science Foundation (U.S.) (Grant No. DMR- 0706078) Natural Sciences and Engineering Research Council of Canada QuantumWorks Corporation Canadian Institute for Advanced Research 2010-12-22T15:35:07Z 2010-12-22T15:35:07Z 2010-09 2010-10 Article http://purl.org/eprint/type/JournalArticle 1098-0121 1550-235X http://hdl.handle.net/1721.1/60355 Chen, Xie et al. “Tensor product representation of a topological ordered phase: Necessary symmetry conditions.” Physical Review B 82.16 (2010): 165119. © 2010 The American Physical Society. https://orcid.org/0000-0001-7296-523X https://orcid.org/0000-0002-5874-581X en_US http://dx.doi.org/10.1103/PhysRevB.82.165119 Physical Review B Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf American Physical Society APS |
| spellingShingle | Chuang, Isaac L. Chen, Xie Wen, Xiao-Gang Zeng, Bei Gu, Zheng-Cheng Tensor product representation of a topological ordered phase: Necessary symmetry conditions |
| title | Tensor product representation of a topological ordered phase: Necessary symmetry conditions |
| title_full | Tensor product representation of a topological ordered phase: Necessary symmetry conditions |
| title_fullStr | Tensor product representation of a topological ordered phase: Necessary symmetry conditions |
| title_full_unstemmed | Tensor product representation of a topological ordered phase: Necessary symmetry conditions |
| title_short | Tensor product representation of a topological ordered phase: Necessary symmetry conditions |
| title_sort | tensor product representation of a topological ordered phase necessary symmetry conditions |
| url | http://hdl.handle.net/1721.1/60355 https://orcid.org/0000-0001-7296-523X https://orcid.org/0000-0002-5874-581X |
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