The spin-spin correlation function in the ground state of some two-dimensional fully frustrated spin models are investigated. The correlations are long ranged and decay as $\frac{1}{{r}^{\ensuremath{\eta}}}$, where $r$ is the distance between the two spins and $\ensuremath{\eta}=\frac{1}{2}$. This result is true for both the square and triangular fully frustrated lattices, thus suggesting that these models form a universality class. It is shown that the fully frustrated Ising model on a square lattice can be mapped exactly into a special Baxter model. The spin-spin correlation function of this Baxter model can be calculated exactly. The ground state of the fully frustrated square Ising model corresponds to the free fermion point of the $F$ model and therefore, through duality, to the decoupling point of the critical Baxter line. The relationship between the $F$ model above the critical temperature and the Gaussian model above the multicritical point is discussed.The spin-spin correlation function in the ground state of some two-dimensional fully frustrated spin models are investigated. The correlations are long ranged and decay as $\frac{1}{{r}^{\ensuremath{\eta}}}$, where $r$ is the distance between the two spins and $\ensuremath{\eta}=\frac{1}{2}$. This result is true for both the square and triangular fully frustrated lattices, thus suggesting that these models form a universality class. It is shown that the fully frustrated Ising model on a square lattice can be mapped exactly into a special Baxter model. The spin-spin correlation function of this Baxter model can be calculated exactly. The ground state of the fully frustrated square Ising model corresponds to the free fermion point of the $F$ model and therefore, through duality, to the decoupling point of the critical Baxter line. The relationship between the $F$ model above the critical temperature and the Gaussian model above the multicritical point is discussed.

Physical review. B, Condensed matterNov 1, 198073 citations

Computation of correlation functions of the two dimensional Ising model

Bülent Kutlu, Nevzat AKTEKİN

Physica A Statistical Mechanics and its ApplicationsMay 1, 199512 citations

Simulação Monte Carlo e percolação no modelo XY completamente frustado

Anderson Barbosa Lima

Using Monte Carlo and spin dynamics techniques we have investigated the critical behavior of the classical fully frustrated XY and fully frustrated Planar Rotor models in two dimensions in square lattices of size up to 256. We have found that the models have two transitions a Berezinskii-Kosterliz-Thouless transition at TBKT = 0:3655(5) and an Ising transition at TI = 0:3690(3) for the fully frustrated XY model and a Berezinskii-Kosterliz-Thouless transition at TBKT = 0:4410(5) and an Ising transition at TI = 0:4505(5) for the fully frustrated Planar Rotor model. After this _rst step we have simulated the Z2 (Ising) transition in both fully frustrated Planar Rotator and fully frustrated XY models. We have shown that they can be understood as a percolation problem if we de_ne properly the connectivity of the system. The critical exponent of the mean cluster size, is found as = 2:2(2) close to the two dimensional Ising case, = 2:389. The critical Fisher's exponent is obtained as _ = 1:8(1), close to the two dimensional Ising value _ = 2:055. Also, weobtain the critical Z2 temperatures for both models using the percolation criterion which agree very well with other calculations.Some preliminary calculations was done in the XY model in order to obtain the neutron scattering correlation function S(q; !) and the dispersion relation.Using Monte Carlo and spin dynamics techniques we have investigated the critical behavior of the classical fully frustrated XY and fully frustrated Planar Rotor models in two dimensions in square lattices of size up to 256. We have found that the models have two transitions a Berezinskii-Kosterliz-Thouless transition at TBKT = 0:3655(5) and an Ising transition at TI = 0:3690(3) for the fully frustrated XY model and a Berezinskii-Kosterliz-Thouless transition at TBKT = 0:4410(5) and an Ising transition at TI = 0:4505(5) for the fully frustrated Planar Rotor model. After this _rst step we have simulated the Z2 (Ising) transition in both fully frustrated Planar Rotator and fully frustrated XY models. We have shown that they can be understood as a percolation problem if we de_ne properly the connectivity of the system. The critical exponent of the mean cluster size, is found as = 2:2(2) close to the two dimensional Ising case, = 2:389. The critical Fisher's exponent is obtained as _ = 1:8(1), close to the two dimensional Ising value _ = 2:055. Also, weobtain the critical Z2 temperatures for both models using the percolation criterion which agree very well with other calculations.Some preliminary calculations was done in the XY model in order to obtain the neutron scattering correlation function S(q; !) and the dispersion relation.

Apr 14, 20050 citations

Surface behavior of the fully frustrated two-dimensional Ising model

Ihnsouk Guim, Theodore W. Burkhardt, Tianyou Xue

We consider the fully frustrated Villain or ``odd'' Ising model on a semi-infinite square lattice with surface couplings ${\mathit{J}}_{\mathit{S}}$ and bulk couplings \ifmmode\pm\else\textpm\fi{}J. The zero-temperature properties of this model have an interpretation in terms of dimers. From previous work it is known that at T=0 there is no bulk long-range order and the bulk spin-spin correlation function decays algebraically with critical exponent \ensuremath{\eta}=1/2. We analyze the surface behavior in the T=0 transition. For ${\mathit{J}}_{\mathit{S}}$ \ensuremath{\ne}1/2J the surface spins do order at T=0, and the connected pair-correlation function decays as ${\mathit{r}}^{\mathrm{\ensuremath{-}}2}$ parallel to the surface. For ${\mathit{J}}_{\mathit{S}}$ =1/2J there is no long-range surface order at T=0, and the pair-correlation function of the surface spins decays as ${\mathit{r}}^{\mathrm{\ensuremath{-}}1}$ and ${\mathit{r}}^{\mathrm{\ensuremath{-}}3}$ for r even and odd, respectively. We also calculate the free energy, the correlation length, and the profile of the energy density in strips of finite width and compare the results with predictions of conformal invariance for ordinary critical systems.We consider the fully frustrated Villain or ``odd'' Ising model on a semi-infinite square lattice with surface couplings ${\mathit{J}}_{\mathit{S}}$ and bulk couplings \ifmmode\pm\else\textpm\fi{}J. The zero-temperature properties of this model have an interpretation in terms of dimers. From previous work it is known that at T=0 there is no bulk long-range order and the bulk spin-spin correlation function decays algebraically with critical exponent \ensuremath{\eta}=1/2. We analyze the surface behavior in the T=0 transition. For ${\mathit{J}}_{\mathit{S}}$\ensuremath{\ne}1/2J the surface spins do order at T=0, and the connected pair-correlation function decays as ${\mathit{r}}^{\mathrm{\ensuremath{-}}2}$ parallel to the surface. For ${\mathit{J}}_{\mathit{S}}$=1/2J there is no long-range surface order at T=0, and the pair-correlation function of the surface spins decays as ${\mathit{r}}^{\mathrm{\ensuremath{-}}1}$ and ${\mathit{r}}^{\mathrm{\ensuremath{-}}3}$ for r even and odd, respectively. We also calculate the free energy, the correlation length, and the profile of the energy density in strips of finite width and compare the results with predictions of conformal invariance for ordinary critical systems.

Physical review. B, Condensed matterDec 1, 19902 citations

Correlation Function of the Two-Dimensional Ising Model on a Finite Lattice: II

A. I. Bugriĭ, O. Lisovyy

Theoretical and Mathematical PhysicsJun 25, 200443 citations

r

is the distance between the two spins and $\ensuremath{\eta}=\frac{1}{2}$. This result is true for both the square and triangular fully frustrated lattices, thus suggesting that these models form a universality class. It is shown that the fully frustrated Ising model on a square lattice can be mapped exactly into a special Baxter model. The spin-spin correlation function of this Baxter model can be calculated exactly. The ground state of the fully frustrated square Ising model corresponds to the free fermion point of the

F

model and therefore, through duality, to the decoupling point of the critical Baxter line. The relationship between the

F

model above the critical temperature and the Gaussian model above the multicritical point is discussed.

We consider the fully frustrated Villain or ``odd'' Ising model on a semi-infinite square lattice with surface couplings

{\mathit{J}}_{\mathit{S}}

and bulk couplings \ifmmode\pm\else\textpm\fi{}J. The zero-temperature properties of this model have an interpretation in terms of dimers. From previous work it is known that at T=0 there is no bulk long-range order and the bulk spin-spin correlation function decays algebraically with critical exponent \ensuremath{\eta}=1/2. We analyze the surface behavior in the T=0 transition. For

{\mathit{J}}_{\mathit{S}}

\ensuremath{\ne}1/2J the surface spins do order at T=0, and the connected pair-correlation function decays as ${\mathit{r}}^{\mathrm{\ensuremath{-}}2}$ parallel to the surface. For

{\mathit{J}}_{\mathit{S}}

=1/2J there is no long-range surface order at T=0, and the pair-correlation function of the surface spins decays as ${\mathit{r}}^{\mathrm{\ensuremath{-}}1}$ and ${\mathit{r}}^{\mathrm{\ensuremath{-}}3}$ for r even and odd, respectively. We also calculate the free energy, the correlation length, and the profile of the energy density in strips of finite width and compare the results with predictions of conformal invariance for ordinary critical systems.