# Integrable modification of the critical Chalker-Coddington network model

## Abstract

We consider the Chalker-Coddington network model for the integer quantum Hall effect, and examine the possibility of solving it exactly. In the supersymmetric path integral framework, we introduce a truncation procedure, leading to a series of well-defined two-dimensional loop models with two loop flavors. In the phase diagram of the first-order truncated model, we identify four integrable branches related to the dilute Birman-Wenzl-Murakami braid-monoid algebra and parameterized by the loop fugacity n. In the continuum limit, two of these branches (1,2) are described by a pair of decoupled copies of a Coulomb-gas theory, whereas the other two branches (3,4) couple the two loop flavors, and relate to an SU(2){sub r}xSU(2){sub r}/SU(2){sub 2r} Wess-Zumino-Witten (WZW) coset model for the particular values n=-2cos[{pi}/(r+2)], where r is a positive integer. The truncated Chalker-Coddington model is the n=0 point of branch 4. By numerical diagonalization, we find that its universality class is neither an analytic continuation of the WZW coset nor the universality class of the original Chalker-Coddington model. It constitutes rather an integrable, critical approximation to the latter.

- Authors:

- Section Mathematiques, Universite de Geneve 2-4 rue du Lievre, CP 64, SZ-1211 Geneve 4 (Switzerland)
- (United States)
- (United Kingdom) and All Souls College, Oxford (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 21596895

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review. B, Condensed Matter and Materials Physics

- Additional Journal Information:
- Journal Volume: 84; Journal Issue: 14; Other Information: DOI: 10.1103/PhysRevB.84.144201; (c) 2011 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1098-0121

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; COMPUTERIZED SIMULATION; FLAVOR MODEL; HALL EFFECT; MODIFICATIONS; PHASE DIAGRAMS; SU-2 GROUPS; SUPERSYMMETRY; TWO-DIMENSIONAL CALCULATIONS; CALCULATION METHODS; COMPOSITE MODELS; DIAGRAMS; INFORMATION; LIE GROUPS; MATHEMATICAL MODELS; PARTICLE MODELS; QUARK MODEL; SIMULATION; SU GROUPS; SYMMETRY; SYMMETRY GROUPS

### Citation Formats

```
Ikhlef, Yacine, Fendley, Paul, Cardy, John, Microsoft Research, Station Q, University of California Santa Barbara, California 93106, USA and Department of Physics, University of Virginia Charlottesville, Virginia 22904-4714, and Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, OX1 3NP.
```*Integrable modification of the critical Chalker-Coddington network model*. United States: N. p., 2011.
Web. doi:10.1103/PHYSREVB.84.144201.

```
Ikhlef, Yacine, Fendley, Paul, Cardy, John, Microsoft Research, Station Q, University of California Santa Barbara, California 93106, USA and Department of Physics, University of Virginia Charlottesville, Virginia 22904-4714, & Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, OX1 3NP.
```*Integrable modification of the critical Chalker-Coddington network model*. United States. doi:10.1103/PHYSREVB.84.144201.

```
Ikhlef, Yacine, Fendley, Paul, Cardy, John, Microsoft Research, Station Q, University of California Santa Barbara, California 93106, USA and Department of Physics, University of Virginia Charlottesville, Virginia 22904-4714, and Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, OX1 3NP. Sat .
"Integrable modification of the critical Chalker-Coddington network model". United States. doi:10.1103/PHYSREVB.84.144201.
```

```
@article{osti_21596895,
```

title = {Integrable modification of the critical Chalker-Coddington network model},

author = {Ikhlef, Yacine and Fendley, Paul and Cardy, John and Microsoft Research, Station Q, University of California Santa Barbara, California 93106, USA and Department of Physics, University of Virginia Charlottesville, Virginia 22904-4714 and Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, OX1 3NP},

abstractNote = {We consider the Chalker-Coddington network model for the integer quantum Hall effect, and examine the possibility of solving it exactly. In the supersymmetric path integral framework, we introduce a truncation procedure, leading to a series of well-defined two-dimensional loop models with two loop flavors. In the phase diagram of the first-order truncated model, we identify four integrable branches related to the dilute Birman-Wenzl-Murakami braid-monoid algebra and parameterized by the loop fugacity n. In the continuum limit, two of these branches (1,2) are described by a pair of decoupled copies of a Coulomb-gas theory, whereas the other two branches (3,4) couple the two loop flavors, and relate to an SU(2){sub r}xSU(2){sub r}/SU(2){sub 2r} Wess-Zumino-Witten (WZW) coset model for the particular values n=-2cos[{pi}/(r+2)], where r is a positive integer. The truncated Chalker-Coddington model is the n=0 point of branch 4. By numerical diagonalization, we find that its universality class is neither an analytic continuation of the WZW coset nor the universality class of the original Chalker-Coddington model. It constitutes rather an integrable, critical approximation to the latter.},

doi = {10.1103/PHYSREVB.84.144201},

journal = {Physical Review. B, Condensed Matter and Materials Physics},

issn = {1098-0121},

number = 14,

volume = 84,

place = {United States},

year = {2011},

month = {10}

}