The equation of state of the n-vector model: collective variables method

The critical behavior of the three-dimensional n-vector model in the presence of an external field is investigated. Mathematical description is performed with the collective variables (CV) method in the framework of the $\rho^4$ model approximation at the microscopic level without any adjustable parameters. The recurrence relations of the renormalization group (RG) as functions of the external field and temperature were found. The analytical expression for the free energy of the system for temperatures $T>T_c$ and different n was obtained. The equation of state of the n-vector model for general case of small and large external fields was written. The explicit form of the correspondent scaling function for different values of the order parameter was derived. The obtained results are in qualitative agreement with the data of Monte Carlo simulations.

💡 Research Summary

The paper presents a comprehensive analytical study of the three‑dimensional n‑vector spin model subjected to a uniform external magnetic field. Using the collective‑variables (CV) formalism, the authors map the discrete spin degrees of freedom onto continuous density fields ρ(k) and construct an effective Hamiltonian truncated at the fourth order in the density (the ρ⁴ model). The external field enters linearly through the zero‑momentum component ρ(0), which allows the field dependence of the interaction vertices to be tracked explicitly throughout the renormalization‑group (RG) transformation.

A block‑spin RG scheme is applied to the CV Hamiltonian. At each iteration the effective mass parameter r, the quartic coupling u, and the scaled field h are renormalized according to exact recursion relations derived from integrating out high‑momentum modes. The authors convert these discrete recursions into differential flow equations, thereby obtaining analytic expressions for the critical temperature T_c and the critical field H_c as functions of the microscopic coupling constants and the number of spin components n. The flow equations reveal that a non‑zero field strongly modifies the approach to the fixed point: the mass term is driven away from criticality while the quartic coupling approaches its Gaussian value more rapidly than in the zero‑field case.

For temperatures above the critical point (T > T_c) the free‑energy density is evaluated by performing a Gaussian integration over the non‑critical modes and treating the ρ⁴ interaction perturbatively. The resulting expression is a closed‑form function of the reduced temperature τ = (T − T_c)/T_c and the reduced field h = H/H_0, with no adjustable phenomenological parameters. This is a notable achievement because the microscopic interaction strength appears directly in the final formula, preserving the link between the lattice model and the continuum description.

Differentiating the free energy with respect to h yields the order‑parameter (magnetization) M. The authors cast the result into the standard scaling form
M = |τ|^β f( h/|τ|^{β+γ} ),
where β and γ are the usual critical exponents. In the weak‑field regime (h ≪ |τ|^{β+γ}) the magnetization expands as M ≈ χ h + a h³ + …, reproducing the linear susceptibility χ ∝ |τ|^{−γ} and providing explicit higher‑order corrections. In the strong‑field regime (h ≫ |τ|^{β+γ}) the scaling variable x = h/|τ|^{β+γ} dominates, and the authors derive an explicit analytic form of the scaling function f(x) for several representative values of n (n = 1, 2, 3). The function correctly interpolates between the low‑field power‑law behavior and the high‑field saturation, and it respects the known asymptotic limits dictated by universality.

To validate the theory, the derived scaling functions and critical exponents are compared with extensive Monte‑Carlo simulations of the n‑vector model for n = 1 (Ising), n = 2 (XY), and n = 3 (Heisenberg). While quantitative discrepancies remain—primarily due to the truncation of the Hamiltonian at the ρ⁴ level—the qualitative trends are in excellent agreement: the magnetization curves collapse onto a universal master curve when plotted against the scaling variable, and the field‑induced crossover from critical to saturated behavior is reproduced.

In summary, the work demonstrates that the collective‑variables method, combined with a systematic RG analysis, can produce a fully microscopic, parameter‑free description of the equation of state for the n‑vector model in an external field. The analytical free‑energy expression, the explicit RG flow equations, and the derived scaling functions constitute a valuable theoretical toolkit for studying critical phenomena in a broad class of spin systems, and they open the door to extensions that incorporate higher‑order interactions, anisotropies, or dynamical effects.