Integrals of motion for one-dimensional Anderson localized systems
Anderson localization is known to be inevitable in one-dimension for generic disordered models. Since localization leads to Poissonian energy level statistics, we ask if localized systems possess ‘additional’ integrals of motion as well, so as to enhance the analogy with quantum integrable systems....
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2016-01-01
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| Series: | New Journal of Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1088/1367-2630/18/3/033010 |
| _version_ | 1797751064774049792 |
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| author | Ranjan Modak Subroto Mukerjee Emil A Yuzbashyan B Sriram Shastry |
| author_facet | Ranjan Modak Subroto Mukerjee Emil A Yuzbashyan B Sriram Shastry |
| author_sort | Ranjan Modak |
| collection | DOAJ |
| description | Anderson localization is known to be inevitable in one-dimension for generic disordered models. Since localization leads to Poissonian energy level statistics, we ask if localized systems possess ‘additional’ integrals of motion as well, so as to enhance the analogy with quantum integrable systems. We answer this in the affirmative in the present work. We construct a set of nontrivial integrals of motion for Anderson localized models, in terms of the original creation and annihilation operators. These are found as a power series in the hopping parameter. The recently found Type-1 Hamiltonians, which are known to be quantum integrable in a precise sense, motivate our construction. We note that these models can be viewed as disordered electron models with infinite-range hopping, where a similar series truncates at the linear order. We show that despite the infinite range hopping, all states but one are localized. We also study the conservation laws for the disorder free Aubry–Andre model, where the states are either localized or extended, depending on the strength of a coupling constant. We formulate a specific procedure for averaging over disorder, in order to examine the convergence of the power series. Using this procedure in the Aubry–Andre model, we show that integrals of motion given by our construction are well-defined in localized phase, but not so in the extended phase. Finally, we also obtain the integrals of motion for a model with interactions to lowest order in the interaction. |
| first_indexed | 2024-03-12T16:42:00Z |
| format | Article |
| id | doaj.art-eda1c540a2214df5adc3c9d87967dc9d |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:42:00Z |
| publishDate | 2016-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-eda1c540a2214df5adc3c9d87967dc9d2023-08-08T14:28:11ZengIOP PublishingNew Journal of Physics1367-26302016-01-0118303301010.1088/1367-2630/18/3/033010Integrals of motion for one-dimensional Anderson localized systemsRanjan Modak0Subroto Mukerjee1Emil A Yuzbashyan2B Sriram Shastry3Department of Physics, Indian Institute of Science , Bangalore 560 012, IndiaDepartment of Physics, Indian Institute of Science , Bangalore 560 012, India; Centre for Quantum Information and Quantum Computing, Indian Institute of Science , Bangalore 560 012, IndiaCenter for Materials Theory, Rutgers University , Piscataway, NJ 08854, USAPhysics Department, University of California , Santa Cruz, CA 95064, USAAnderson localization is known to be inevitable in one-dimension for generic disordered models. Since localization leads to Poissonian energy level statistics, we ask if localized systems possess ‘additional’ integrals of motion as well, so as to enhance the analogy with quantum integrable systems. We answer this in the affirmative in the present work. We construct a set of nontrivial integrals of motion for Anderson localized models, in terms of the original creation and annihilation operators. These are found as a power series in the hopping parameter. The recently found Type-1 Hamiltonians, which are known to be quantum integrable in a precise sense, motivate our construction. We note that these models can be viewed as disordered electron models with infinite-range hopping, where a similar series truncates at the linear order. We show that despite the infinite range hopping, all states but one are localized. We also study the conservation laws for the disorder free Aubry–Andre model, where the states are either localized or extended, depending on the strength of a coupling constant. We formulate a specific procedure for averaging over disorder, in order to examine the convergence of the power series. Using this procedure in the Aubry–Andre model, we show that integrals of motion given by our construction are well-defined in localized phase, but not so in the extended phase. Finally, we also obtain the integrals of motion for a model with interactions to lowest order in the interaction.https://doi.org/10.1088/1367-2630/18/3/033010Anderson localizationintegrals of motionlocalization–delocalization transition02.30.Ik05.30.-d05.45.Mt |
| spellingShingle | Ranjan Modak Subroto Mukerjee Emil A Yuzbashyan B Sriram Shastry Integrals of motion for one-dimensional Anderson localized systems New Journal of Physics Anderson localization integrals of motion localization–delocalization transition 02.30.Ik 05.30.-d 05.45.Mt |
| title | Integrals of motion for one-dimensional Anderson localized systems |
| title_full | Integrals of motion for one-dimensional Anderson localized systems |
| title_fullStr | Integrals of motion for one-dimensional Anderson localized systems |
| title_full_unstemmed | Integrals of motion for one-dimensional Anderson localized systems |
| title_short | Integrals of motion for one-dimensional Anderson localized systems |
| title_sort | integrals of motion for one dimensional anderson localized systems |
| topic | Anderson localization integrals of motion localization–delocalization transition 02.30.Ik 05.30.-d 05.45.Mt |
| url | https://doi.org/10.1088/1367-2630/18/3/033010 |
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