An antiferromagnetic classical XY model with the nearest-neighbor interaction 2 J and the next-nearest-neighbor interaction δ J on a one-dimensional lattice is investigated in the low temperature limit. We obtain the asymptotic behaviors at long distances, of the two-spin correlation function for δ>1/2 and of the chirality-chirality correlation function for δ=1. The behavior of the latter correlation function is shown to be similar to the one of the two-spin correlation function for the Ising model with the nearest-neighbor interaction J /2 on a one-dimensional lattice.An antiferromagnetic classical XY model with the nearest-neighbor interaction 2 J and the next-nearest-neighbor interaction δ J on a one-dimensional lattice is investigated in the low temperature limit. We obtain the asymptotic behaviors at long distances, of the two-spin correlation function for δ>1/2 and of the chirality-chirality correlation function for δ=1. The behavior of the latter correlation function is shown to be similar to the one of the two-spin correlation function for the Ising model with the nearest-neighbor interaction J /2 on a one-dimensional lattice.

Journal of the Physical Society of JapanMar 15, 199014 citations

Two-tag correlation functions in one-dimensional lattice gases

Satya N. Majumdar, Mustansir Barma

Physica A Statistical Mechanics and its ApplicationsSep 1, 199130 citations

Decay of the direct correlation function in linear lattice systems

M. Robert

The direct correlation function of the one-dimensional lattice-gas (Ising) model with nearest-neighbor and next-nearest-neighbor interactions is calculated exactly. It is shown that, depending on the strengths and signs of the coupling constants, the direct correlation function can either have a finite range equal to that of the interactions, or decay exponentially in a monotonic or oscillatory fashion. At the critical point, which occurs at zero temperature, the direct correlation function is found in all cases to have exactly the range of the interactions, while its values become unbounded; this is in contrast to earlier nonrigorous results. The second moment of the direct correlation function is found to diverge at the critical point, in the neighborhood of which it becomes proportional to the correlation length of the density (spin) fluctuations, confirming a prediction of scaling theory in one dimension.The direct correlation function of the one-dimensional lattice-gas (Ising) model with nearest-neighbor and next-nearest-neighbor interactions is calculated exactly. It is shown that, depending on the strengths and signs of the coupling constants, the direct correlation function can either have a finite range equal to that of the interactions, or decay exponentially in a monotonic or oscillatory fashion. At the critical point, which occurs at zero temperature, the direct correlation function is found in all cases to have exactly the range of the interactions, while its values become unbounded; this is in contrast to earlier nonrigorous results. The second moment of the direct correlation function is found to diverge at the critical point, in the neighborhood of which it becomes proportional to the correlation length of the density (spin) fluctuations, confirming a prediction of scaling theory in one dimension.

Physical Review AMay 1, 19847 citations

n-Particle correlation functions of an interacting electron gas

M. V. Vavrukh

Theoretical and Mathematical PhysicsMar 1, 19825 citations

Correlation functions in one-dimensional systems with strong interaction

K. B. Efetov, A.I. Lapkin

Aug 1, 197517 citations

Partial Test of the Universality Hypothesis: The Case of Next-Nearest-Neighbor Interactions

Gerald Paul, Eugene Stanley

High-temperature series expansions are used to examine the dependence of critical-point exponents upon the presence of second-neighbor interactions. We consider the Hamiltonian ${\mathcal{H}}_{\mathrm{nnn}}=\ensuremath{-}{J}_{1}\ensuremath{\Sigma}\stackrel{\mathrm{nn}}{〈\mathrm{ij}〉}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}^{(D)}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{j}^{(D)}\ensuremath{-}{J}_{2}\ensuremath{\Sigma}\stackrel{\mathrm{nnn}}{〈\mathrm{ij}〉}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}^{(D)}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{j}^{(D)},$ where the first and second sums are over pairs of nearest-neighbor (nn) and next-nearest-neighbor (nnn) sites, and where the spins ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}^{(D)}$ are $D$ -dimensional unit vectors. The two-spin correlation function, ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, is calculated to tenth, ninth, and eighth order in $\frac{1}{{k}_{B}T}$ for the Ising ( $D=1$ ), classical-planar ( $D=2$ ), and classical-Heisenberg ( $D=3$ ) models, respectively, for various values of the parameter ${R}^{\ensuremath{'}}\ensuremath{\equiv}\frac{{J}_{2}}{{J}_{1}}$ and for various cubic lattices (fcc, bcc, and sample cubic). These represent the first series expansions of the spin correlation function for nnn interactions. From ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ we obtain series for the specific heat, susceptibility, and second moment. Analysis of these series and detailed comparisons with the exactly soluble spherical model ($D=\ensuremath{\infty}$) lead us to conclude that the exponents $\ensuremath{\gamma}$ (susceptibility) and $\ensuremath{\nu}$ (correlation length) may be independent of ${R}^{\ensuremath{'}}$; this suggestion is consistent with the universality hypothesis.High-temperature series expansions are used to examine the dependence of critical-point exponents upon the presence of second-neighbor interactions. We consider the Hamiltonian ${\mathcal{H}}_{\mathrm{nnn}}=\ensuremath{-}{J}_{1}\ensuremath{\Sigma}\stackrel{\mathrm{nn}}{〈\mathrm{ij}〉}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}^{(D)}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{j}^{(D)}\ensuremath{-}{J}_{2}\ensuremath{\Sigma}\stackrel{\mathrm{nnn}}{〈\mathrm{ij}〉}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}^{(D)}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{j}^{(D)},$ where the first and second sums are over pairs of nearest-neighbor (nn) and next-nearest-neighbor (nnn) sites, and where the spins ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}^{(D)}$ are $D$-dimensional unit vectors. The two-spin correlation function, ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, is calculated to tenth, ninth, and eighth order in $\frac{1}{{k}_{B}T}$ for the Ising ($D=1$), classical-planar ($D=2$), and classical-Heisenberg ($D=3$) models, respectively, for various values of the parameter ${R}^{\ensuremath{'}}\ensuremath{\equiv}\frac{{J}_{2}}{{J}_{1}}$ and for various cubic lattices (fcc, bcc, and sample cubic). These represent the first series expansions of the spin correlation function for nnn interactions. From ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ we obtain series for the specific heat, susceptibility, and second moment. Analysis of these series and detailed comparisons with the exactly soluble spherical model ($D=\ensuremath{\infty}$) lead us to conclude that the exponents $\ensuremath{\gamma}$ (susceptibility) and $\ensuremath{\nu}$ (correlation length) may be independent of ${R}^{\ensuremath{'}}$; this suggestion is consistent with the universality hypothesis.

Physical review. B, Solid stateMay 1, 197230 citations

D

-dimensional unit vectors. The two-spin correlation function, ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, is calculated to tenth, ninth, and eighth order in

\frac{1}{{k}_{B}T}

for the Ising (

D=1

), classical-planar (

D=2

), and classical-Heisenberg (

D=3

) models, respectively, for various values of the parameter ${R}^{\ensuremath{'}}\ensuremath{\equiv}\frac{{J}_{2}}{{J}_{1}}$ and for various cubic lattices (fcc, bcc, and sample cubic). These represent the first series expansions of the spin correlation function for nnn interactions. From ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ we obtain series for the specific heat, susceptibility, and second moment. Analysis of these series and detailed comparisons with the exactly soluble spherical model ($D=\ensuremath{\infty}$) lead us to conclude that the exponents $\ensuremath{\gamma}$ (susceptibility) and $\ensuremath{\nu}$ (correlation length) may be independent of ${R}^{\ensuremath{'}}$; this suggestion is consistent with the universality hypothesis.