### Abstract

The zero-temperature, classical XY model on an L×L square lattice is studied by exploring the distribution ΦL(y) of its centered and normalized magnetization y in the large-L limit. An integral representation of the cumulant generating function, known from earlier works, is used for the numerical evaluation of ΦL(y), and the limit distribution ΦL→∞(y)=Φ0(y) is obtained with high precision. The two leading finite-size corrections ΦL(y)-Φ0(y)≈a1(L)Φ1(y)+a2(L)Φ2(y) are also extracted both from numerics and from analytic calculations. We find that the amplitude a1(L) scales as ln(L/L0)/L2 and the shape correction function Φ1(y) can be expressed through the low-order derivatives of the limit distribution, Φ1(y)=[yΦ0(y)+Φ0′(y)]′. Thus, Φ1(y) carries the same universal features as the limit distribution and can be used for consistency checks of universality claims based on finite-size systems. The second finite-size correction has an amplitude a2(L)∝1/L2 and one finds that a2Φ2(y)a1Φ1(y) already for small system size (L>10). We illustrate the feasibility of observing the calculated finite-size corrections by performing simulations of the XY model at low temperatures, including T=0.

Original language | English |
---|---|

Article number | 022145 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 94 |

Issue number | 2 |

DOIs | |

Publication status | Published - Aug 29 2016 |

### Fingerprint

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics

### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*94*(2), [022145]. https://doi.org/10.1103/PhysRevE.94.022145

**Finite-size corrections to scaling of the magnetization distribution in the two-dimensional XY model at zero temperature.** / Palma, G.; Niedermayer, F.; Rácz, Z.; Riveros, A.; Zambrano, D.

Research output: Contribution to journal › Article

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 94, no. 2, 022145. https://doi.org/10.1103/PhysRevE.94.022145

}

TY - JOUR

T1 - Finite-size corrections to scaling of the magnetization distribution in the two-dimensional XY model at zero temperature

AU - Palma, G.

AU - Niedermayer, F.

AU - Rácz, Z.

AU - Riveros, A.

AU - Zambrano, D.

PY - 2016/8/29

Y1 - 2016/8/29

N2 - The zero-temperature, classical XY model on an L×L square lattice is studied by exploring the distribution ΦL(y) of its centered and normalized magnetization y in the large-L limit. An integral representation of the cumulant generating function, known from earlier works, is used for the numerical evaluation of ΦL(y), and the limit distribution ΦL→∞(y)=Φ0(y) is obtained with high precision. The two leading finite-size corrections ΦL(y)-Φ0(y)≈a1(L)Φ1(y)+a2(L)Φ2(y) are also extracted both from numerics and from analytic calculations. We find that the amplitude a1(L) scales as ln(L/L0)/L2 and the shape correction function Φ1(y) can be expressed through the low-order derivatives of the limit distribution, Φ1(y)=[yΦ0(y)+Φ0′(y)]′. Thus, Φ1(y) carries the same universal features as the limit distribution and can be used for consistency checks of universality claims based on finite-size systems. The second finite-size correction has an amplitude a2(L)∝1/L2 and one finds that a2Φ2(y)a1Φ1(y) already for small system size (L>10). We illustrate the feasibility of observing the calculated finite-size corrections by performing simulations of the XY model at low temperatures, including T=0.

AB - The zero-temperature, classical XY model on an L×L square lattice is studied by exploring the distribution ΦL(y) of its centered and normalized magnetization y in the large-L limit. An integral representation of the cumulant generating function, known from earlier works, is used for the numerical evaluation of ΦL(y), and the limit distribution ΦL→∞(y)=Φ0(y) is obtained with high precision. The two leading finite-size corrections ΦL(y)-Φ0(y)≈a1(L)Φ1(y)+a2(L)Φ2(y) are also extracted both from numerics and from analytic calculations. We find that the amplitude a1(L) scales as ln(L/L0)/L2 and the shape correction function Φ1(y) can be expressed through the low-order derivatives of the limit distribution, Φ1(y)=[yΦ0(y)+Φ0′(y)]′. Thus, Φ1(y) carries the same universal features as the limit distribution and can be used for consistency checks of universality claims based on finite-size systems. The second finite-size correction has an amplitude a2(L)∝1/L2 and one finds that a2Φ2(y)a1Φ1(y) already for small system size (L>10). We illustrate the feasibility of observing the calculated finite-size corrections by performing simulations of the XY model at low temperatures, including T=0.

UR - http://www.scopus.com/inward/record.url?scp=84989344940&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84989344940&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.94.022145

DO - 10.1103/PhysRevE.94.022145

M3 - Article

AN - SCOPUS:84989344940

VL - 94

JO - Physical review. E

JF - Physical review. E

SN - 2470-0045

IS - 2

M1 - 022145

ER -