## Integrals of motion for one-dimensional Anderson localized systems

## Abstract

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. Weanswer 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.Wenote that these models can be viewed as disordered electron models with infinite-range hopping, where a similar series truncates at the linear order.Weshow that despite the infinite range hopping, all states but one are localized.Wealso 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.Weformulate 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-definedmore »

- Authors:

- Indian Institute of Science, Bangalore (India)
- Rutgers Univ., Piscataway, NJ (United States)
- Univ. of California, Santa Cruz, CA (United States)

- Publication Date:

- Research Org.:
- Univ. of California, Santa Cruz, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1255231

- Grant/Contract Number:
- FG02-06ER46319

- Resource Type:
- Accepted Manuscript

- Journal Name:
- New Journal of Physics

- Additional Journal Information:
- Journal Volume: 18; Journal Issue: 3; Journal ID: ISSN 1367-2630

- Publisher:
- IOP Publishing

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Anderson localization; integrals of motion; localization–delocalization transition

### Citation Formats

```
Modak, Ranjan, Mukerjee, Subroto, Yuzbashyan, Emil A., and Shastry, B. Sriram. Integrals of motion for one-dimensional Anderson localized systems. United States: N. p., 2016.
Web. doi:10.1088/1367-2630/18/3/033010.
```

```
Modak, Ranjan, Mukerjee, Subroto, Yuzbashyan, Emil A., & Shastry, B. Sriram. Integrals of motion for one-dimensional Anderson localized systems. United States. doi:10.1088/1367-2630/18/3/033010.
```

```
Modak, Ranjan, Mukerjee, Subroto, Yuzbashyan, Emil A., and Shastry, B. Sriram. Wed .
"Integrals of motion for one-dimensional Anderson localized systems". United States. doi:10.1088/1367-2630/18/3/033010. https://www.osti.gov/servlets/purl/1255231.
```

```
@article{osti_1255231,
```

title = {Integrals of motion for one-dimensional Anderson localized systems},

author = {Modak, Ranjan and Mukerjee, Subroto and Yuzbashyan, Emil A. and Shastry, B. Sriram},

abstractNote = {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. Weanswer 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.Wenote that these models can be viewed as disordered electron models with infinite-range hopping, where a similar series truncates at the linear order.Weshow that despite the infinite range hopping, all states but one are localized.Wealso 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.Weformulate 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. Lastly, we also obtain the integrals of motion for a model with interactions to lowest order in the interaction.},

doi = {10.1088/1367-2630/18/3/033010},

journal = {New Journal of Physics},

number = 3,

volume = 18,

place = {United States},

year = {2016},

month = {3}

}

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