# Small instantons in CP{sup 1} and CP{sup 2} sigma models

## Abstract

The anomalous scaling behavior of the topological susceptibility {chi}{sub t} in two-dimensional CP{sup N-1} sigma models for N{<=}3 is studied using the overlap Dirac operator construction of the lattice topological charge density. The divergence of {chi}{sub t} in these models is traced to the presence of small instantons with a radius of order a(=lattice spacing), which are directly observed on the lattice. The observation of these small instantons provides detailed confirmation of Luescher's argument that such short-distance excitations, with quantized topological charge, should be the dominant topological fluctuations in CP{sup 1} and CP{sup 2}, leading to a divergent topological susceptibility in the continuum limit. For the CP{sup N-1} models with N>3 the topological susceptibility is observed to scale properly with the mass gap. These larger N models are not dominated by instantons, but rather by coherent, one-dimensional regions of topological charge which can be interpreted as domain wall or Wilson line excitations and are analogous to D-brane or 'Wilson bag' excitations in QCD. In Lorentz gauge, the small instantons and Wilson line excitations can be described, respectively, in terms of poles and cuts of an analytic gauge potential.

- Authors:

- Department of Physics, University of Virginia, P.O. Box 400714 Charlottesville, Virginia 22901-4714 (United States)

- Publication Date:

- OSTI Identifier:
- 21020210

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 6; Other Information: DOI: 10.1103/PhysRevD.75.065031; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; CHARGE DENSITY; DIRAC OPERATORS; EXCITATION; FLUCTUATIONS; INSTANTONS; ONE-DIMENSIONAL CALCULATIONS; POTENTIALS; QUANTUM CHROMODYNAMICS; SIGMA MODEL; TOPOLOGY; TWO-DIMENSIONAL CALCULATIONS; WILSON LOOP

### Citation Formats

```
Lian Yaogang, and Thacker, H. B.
```*Small instantons in CP{sup 1} and CP{sup 2} sigma models*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVD.75.065031.

```
Lian Yaogang, & Thacker, H. B.
```*Small instantons in CP{sup 1} and CP{sup 2} sigma models*. United States. doi:10.1103/PHYSREVD.75.065031.

```
Lian Yaogang, and Thacker, H. B. Thu .
"Small instantons in CP{sup 1} and CP{sup 2} sigma models". United States.
doi:10.1103/PHYSREVD.75.065031.
```

```
@article{osti_21020210,
```

title = {Small instantons in CP{sup 1} and CP{sup 2} sigma models},

author = {Lian Yaogang and Thacker, H. B.},

abstractNote = {The anomalous scaling behavior of the topological susceptibility {chi}{sub t} in two-dimensional CP{sup N-1} sigma models for N{<=}3 is studied using the overlap Dirac operator construction of the lattice topological charge density. The divergence of {chi}{sub t} in these models is traced to the presence of small instantons with a radius of order a(=lattice spacing), which are directly observed on the lattice. The observation of these small instantons provides detailed confirmation of Luescher's argument that such short-distance excitations, with quantized topological charge, should be the dominant topological fluctuations in CP{sup 1} and CP{sup 2}, leading to a divergent topological susceptibility in the continuum limit. For the CP{sup N-1} models with N>3 the topological susceptibility is observed to scale properly with the mass gap. These larger N models are not dominated by instantons, but rather by coherent, one-dimensional regions of topological charge which can be interpreted as domain wall or Wilson line excitations and are analogous to D-brane or 'Wilson bag' excitations in QCD. In Lorentz gauge, the small instantons and Wilson line excitations can be described, respectively, in terms of poles and cuts of an analytic gauge potential.},

doi = {10.1103/PHYSREVD.75.065031},

journal = {Physical Review. D, Particles Fields},

number = 6,

volume = 75,

place = {United States},

year = {Thu Mar 15 00:00:00 EDT 2007},

month = {Thu Mar 15 00:00:00 EDT 2007}

}