High-temperature expansions of the higher susceptibilities for the Ising model in general dimensiond

Abstract

The high-temperature expansion coefficients of the ordinary and the higher susceptibilities of the spin-1/2 nearest-neighbor Ising model are calculated exactly up to the 20th order for the general d-dimensional (hyper)simple-cubical lattices. These series are analyzed to study the dependence of critical parameters on the lattice dimensionality. Using the general d expression of the ordinary susceptibility, we have more than doubled the length of the existing series expansion of the critical temperature in powers of 1/d.

The Finite-Size Scaling Study of Five-Dimensional Ising Model

“…The dependence of the critical temperatures T χ c (L) obtained from the magnetic susceptibility maxima of the finite-size lattices on linear dimension L is given by the following expression [14,26,[33][34][35][36][37]: agrees with the results obtained previously using different methods [14,26,[37][38][39][40][41][42][43][44][45][46]. For a lattice linear dimension L and very small h at T = T c (L), the order parameter is given by…”

Constraints on Conformal Field Theories in Diverse Dimensions from the Bootstrap Mechanism

“…Similarly, with no more effort, we can apply the same analysis to the Yang Lee edge singularity in four space dimensions, obtaining ∆ ϕ 0.823, ∆ 4 5.71, p ∆ϕ −2.86, p 4,2 0.40 (18) and consequently σ 0.259, to be compared with σ = 0.258(5) of [12]. Figure 3 shows this solution as the common intersection of the zero loci of 3 × 3 minors.…”

“…We can extract from p 2,2 the estimate c −4.53 of the central charge , while its exact value is c = − Inserting this value of ∆ ϕ in (12) we obtain σ 0.076, to be compared with the very accurate estimate [12] σ = 0.077(2), based on very long series expansions of dimer density in powers of the activity in a cubic lattice. Similarly, with no more effort, we can apply the same analysis to the Yang Lee edge singularity in four space dimensions, obtaining ∆ ϕ 0.823, ∆ 4 5.71, p ∆ϕ −2.86, p 4,2 0.40 (18) and consequently σ 0.259, to be compared with σ = 0.258(5) of [12].…”

“…For instance, in the 3d critical Ising model there are two more scalar primary operators of scaling dimension smaller than that of the spin 4 operator, one associated to ϕ 4 with ∆ ϕ 4 = 3.84(4), the other associated to ϕ 6 with ∆ ϕ 6 = 4.67(11) (∆ 4 = 5.0208 (12) , therefore in the latter case the constraint becomes F (∆ ϕ 2 , ∆ ϕ 2 ) = 0, which has a discrete number of solutions. One is at ∆ ϕ 2 1.447.…”

Exact renormalization group for quantum spin systems

“…The calculation of the initial functional Γ 0 [M ] requires the inversion of the Brillouin function which is not possible in closed form [39]. However, we can iteratively [40][41][42][43]. The third column marked O(D −1 ) is the prediction of our analytical formula (15).…”