Green-function theory of the Heisenberg ferromagnet in a magnetic field

We present a second-order Green-function theory of the one- and two-dimensional $S=1∕2$ ferromagnet in a magnetic field based on a decoupling of three-spin operator products, where vertex parameters are introduced and determined by exact relations. The transverse and longitudinal spin correlation functions and thermodynamic properties (magnetization, isothermal magnetic susceptibility, specific heat) are calculated self-consistently at arbitrary temperatures and fields. In addition, exact diagonalizations on finite lattices and, in the one-dimensional case, exact calculations by the Bethe-ansatz method for the quantum transfer matrix are performed. A good agreement of the Green-function theory with the exact data, with recent quantum Monte Carlo results, and with the spin polarization of a $\nu =1$ quantum Hall ferromagnet is obtained. The field dependences of the position and height of the maximum in the temperature dependence of the susceptibility are found to fit well to power laws, which are critically analyzed in relation to the recently discussed behavior in Landau’s theory. As revealed by the spin correlation functions and the specific heat at low fields, our theory provides an improved description of magnetic short-range order as compared with the random phase approximation. In one dimension and at very low fields, two maxima in the temperature dependence of the specific heat are found. The Bethe-ansatz data for the field dependences of the position and height of the low-temperature maximum are described by power laws. At higher fields in one and two dimensions, the temperature of the specific heat maximum linearly increases with the field.

Figure 1

Magnetization of the 1D (a) and 2D (b) Heisenberg ferromagnet in magnetic fields of strengths $h=1.0$, 0.8, 0.6, 0.4, and 0.1, from top to bottom, as obtained by the Green-function theory (solid), the ED (엯), and the Bethe-ansatz method (∎), compared with RPA results (dotted) and QMC data (●, Ref. 4). The inset shows the low-field magnetization of the 1D model at $h=0.05$ and 0.005 from top to bottom.

Figure 2

Magnetization for the $\nu =1$ quantum Hall ferromagnet calculated at $h=0.32$ (solid) in comparison with QMC (●, Ref. 4) and experimental data (◻, Ref. 1).

Figure 3

Isothermal susceptibility of the 1D ferromagnet at low fields (a), $h=0.005$ and 0.05, from top to bottom, and at higher fields (b), $h=0.4$, 1.0, and 2.0, from top to bottom, where the Green-function (solid), ED (엯), Bethe-ansatz (∎), and the RPA results (dotted) are shown. In the inset the 1D zero-field susceptibility is depicted.

Figure 4

Isothermal susceptibility of the 2D ferromagnet at $h=0.4$, 1.0, and 2.0, from top to bottom. The Green-function theory (solid) is compared with ED data (엯) and RPA results (dotted). The inset shows the 2D inverse zero-field susceptibility.

Figure 5

Logarithmic plot of the field dependence of the position (a) and height (b) of the susceptibility maximum obtained by the Green-function theory (▴) and fit to a linear dependence (solid) in comparison with ED (엯) and RPA results (◻).

Figure 6

Nearest-neighbor transverse (a) and longitudinal (b) spin correlation functions of the 1D ferromagnet at the fields $h=0.1$, 0.4, 1.0, and 2.0, from left to right. The Green-function theory (solid) is compared with ED (엯) and RPA results (dotted).

Figure 7

Nearest-neighbor transverse (a) and longitudinal (b) spin correlation functions of the 2D ferromagnet at the fields $h=0.1$, 0.4, 1.0, and 2.0, from left to right. The Green-function theory (solid) is compared with ED (엯) and RPA results (dotted).

Figure 8

Specific heat of the 1D ferromagnet obtained by the Green-function (solid) and Bethe-ansatz method (∎) at low fields (a), $h=0$, 0.005, and 0.03, from bottom to top, with the ED data denoted by ☉, 엯, and ⊗, respectively, and at higher fields (b), $h=0.1$, 1.0, and 2.0, from left to right, in comparison with ED (엯) and RPA results (dotted). For clarity, the Bethe-ansatz data for $h=0.005$ and 0.03 at low temperatures are joined by dotted lines. For low fields the RPA data are not drawn because of the too high maximum [cf. (b)]. The inset exhibits the Bethe-ansatz results for very low fields, $h=0$ to 0.01 in steps of 0.001, from bottom to top.

Figure 9

Specific heat of the 2D ferromagnet at $h=0.1$, 1.0, and 2.0, from left to right, showing the Green-function (solid), ED (엯), and the RPA results (dotted). At $h=0$ the Green-function theory (dashed) is compared with ED data (☉).