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Physical review. B, Condensed matter

Peculiarities of the O (n) model for n<1

P. D. Gujrati

Published Mar 1, 1982• 21 citations
Publisher

Abstract

The $O (n)$ model consisting of $n$-component spins $\stackrel{\ensuremath{\rightarrow}}{S}$ with the constraint ${\stackrel{\ensuremath{\rightarrow}}{S}}^{2}=\ensuremath{\lambda}$ in a magnetic field $h$ is studied. It is shown that mathematically it is possible for the susceptibility to become negative for $n<1$, which implies a violation of convexity properties for $n<1$. In a mean-field approximation, the susceptibility ${\ensuremath{\chi}}_{n}$ and the specific heat ${C}_{n}$ are positive near the critical temperature ${T}_{c}$ for all $n\ensuremath{\ge}0$ in contradiction with the $\ensuremath{\epsilon}$ expansion, but they become negative at very low temperatures for $n<1$. It is also shown that the spontaneous magnetization is not a monotone function of temperature for $n<1$. Our calculation also supports the conclusion drawn by des Cloizeaux that the low-temperature phase of the $O (0)$ model describes the semidilute regime of the polymer system as $h\ensuremath{\rightarrow}0$.

DOI
10.1103/physrevb.25.3381

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The O(n)O (n)O(n) model consisting of nnn-component spins S\ensuremath→\stackrel{\ensuremath{\rightarrow}}{S}S\ensuremath→​ with the constraint S\ensuremath→2=\ensuremathλ{\stackrel{\ensuremath{\rightarrow}}{S}}^{2}=\ensuremath{\lambda}S\ensuremath→​2=\ensuremathλ in a magnetic field hhh is studied. It is shown that mathematically it is possible for the susceptibility to become negative for n<1n<1n<1, which implies a violation of convexity properties for n<1n<1n<1. In a mean-field approximation, the susceptibility \ensuremathχn{\ensuremath{\chi}}_{n}\ensuremathχn​ and the specific heat Cn{C}_{n}Cn​ are positive near the critical temperature Tc{T}_{c}Tc​ for all n\ensuremath≥0n\ensuremath{\ge}0n\ensuremath≥0 in contradiction with the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion, but they become negative at very low temperatures for n<1n<1n<1. It is also shown that the spontaneous magnetization is not a monotone function of temperature for n<1n<1n<1. Our calculation also supports the conclusion drawn by des Cloizeaux that the low-temperature phase of the O(0)O (0)O(0) model describes the semidilute regime of the polymer system as h\ensuremath→0h\ensuremath{\rightarrow}0h\ensuremath→0.
We argue for the logical necessity of the existence of a phase transition in a polymer system which is identical to the n=0 limit of two different magnetic systems, one possessing a continuous symmetry, the other one possessing a discrete symmetry. The phase transition occurs across a curve AC passing through the critical point, as has been argued recently by Gujrati. We accomplish this by demonstrating explicitly a phase transition across AC in the mean-field calculation of the discrete model for n1. We show that the phase below AC in this model has no polymer counterpart, and corresponds to a complete restoration of symmetry among the n components, even though the magnetic field couples to only one of the components. We discuss the nature of the magnetization and the susceptibilities on and near AC in the context of (i) the mean-field calculation with and without fluctuations and (ii) the \ensuremath{\epsilon} expansion. Within the general context of renormalization and scaling, we argue that in the scaling regime near the critical point, the longitudinal susceptibility must be positive for small magnetic fields, and the magnetization must not be less than the spontaneous magnetization, provided there is no transition across AC. Even though this result seems to be in contradiction with the \ensuremath{\epsilon} expansion result, we argue that there is no contradiction, and that our results are consistent with the \ensuremath{\epsilon} expansion results provided the existence of the phase transition is acknowledged.
We investigate low- and high-temperature properties of the ferromagnetic Ising model of infinite-spin. By using the cummulant expansion methods, we obtain low-temperature expansions of the free energy, the specific heat, the spontaneous magnetization, the zero-field susceptibility and so on, and high-temperature expansions of the free energy and the zero-field susceptibility. We compare the results with those obtained by Thompson for the system on the one-dimensional lattice, with those by using Monte Carlo simulations for the systems on the square lattice and on the simple cubic lattice and with those with the aid of the mean-field approximation.
The analysis of a previous study (Allen and Pathria. Can. J. Phys. 67, 952 (1989)) on finite-size effects in systems with O(n) symmetry [Formula: see text], confined to geometry Ld−d′ × ∞d′ (where d and d′ are continuous variables such that 2 < d′ < d < 4) and subjected to periodic boundary conditions, is extended (i) to include the region of first-order phase transition (T < Tc) as well as the region of second-order phase transition [Formula: see text] and (ii) to allow the presence of an external field H > 0. Predictions, involving both amplitudes and exponents, are made on the magnetization m and susceptibility χ in different regimes of the variables T, H, and L. Analytical verification of the predicted results is carried out in the case of the spherical model of ferromagnetism (n = ∞), and complete agreement is found.
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The zero- and low-temperature behaviors of a quasi-one-dimensional organic polymer proposed as a symmetrical periodic Anderson-like chain model, in which the localized f orbitals hybridize with the conduction orbitals at even sites, are investigated by means of many-body Green's function theory. In the absence of magnetic field, the ground state of the system turns out to be ferrimagnetic. The temperature-induced phase diagrams have been explored, where the competition between the Hubbard repulsion U on the localized f orbital and the hybridization strength V makes an important impact on the transition temperature. In a magnetic field, it is found that a 1/3 magnetization plateau appears and two critical fields indicating the insulator-metal transitions at zero temperature emerge, which are closely related to the energy bands. Furthermore, the single-site entanglement entropy is a good indicator of quantum phase transitions. The temperature-field-induced phase diagram has also been attained, wherein the magnetization plateau state, the gapless phase and the spin polarized state are revealed. The temperature dependence of thermodynamic quantities such as the magnetization, susceptibility and specific heat are calculated to characterize the corresponding phases. It is also found that the up-spin and down-spin hole excitations are responsible for the thermodynamic properties.