Frustrated Heisenberg Magnets: A Nonperturbative Approach

Abstract

Frustrated magnets are a notorious example where the usual perturbative methods are in conflict. Using a nonperturbative Wilson-like approach, we get a coherent picture of the physics of Heisenberg frustrated magnets everywhere between d = 2 and d = 4. We recover all known perturbative results in a single framework and find the transition to be weakly first order in d = 3. We compute effective exponents in good agreement with numerical and experimental data.PACS No: 75.10.Hk, 11.10.Hi, 11.15.Tk Understanding the effect of competing interactions in three dimensional classical spin systems is one of the great challenges of condensed matter physics. However, after twenty five years of investigations, the nature of the universality class for the phase transition of the simplest frustrated model, the antiferromagnetic Heisenberg model on a triangular lattice (AFHT model), is still a strongly debated question [1] . Due to frustration, the ground state of the AFHT model is given by a canted configuration -the famous 120 • structure -that implies a matrix-like order parameter [2] and thus, the possibility of a new universality class. Experiments performed on materials supposed to belong to the AFHT universality class display indeed exponents different from those of the standard O(N ) universality class: for VCl These results however call for several comments. First, the exponents violate the scaling relations, at least by two standard deviations. Second, they differ significantly from those obtained by Monte Carlo (MC) simulations performed either directly on the AFHT model (ν ≃ 0.59(1), γ ≃ 1.17(2), β ≃ 0.29(1), α ≃ 0.24(2)), and on models supposed to belong to the same universality class: AFHT with rigid constraints (ν = 0.504(10), γ = 1.074(29), β = 0.221(9), α = 0.488(30)), dihedral (i.e. V 3,2 Stiefel) models (ν ≃ 0.51(1), γ ≃ 1.13(2), β ≃ 0.193(4), α ≃ 0.47(3)). See Ref.[9] for a review, and references therein. Finally, the anomalous dimensions η obtained by means of scaling relations is found to be negative in experiments as well as in MC simulations, a result forbidden by first principles for second order phase transitions [10] . All these results are hardly compatible with the assumption of universality. It has been proposed that the exponents are, in fact, effective exponents characterizing a very weakly first order transition, the so-called "almost second order phase transition [11][12][13] ".From the theoretical point of view the situation is also very unsatisfactory since one does not have a coherent picture of the expected critical behaviour of the AFHT model between two and four dimensions. On the one hand, the weak coupling expansion performed on the suitable Landau-Ginzburg-Wilson (LGW) model in the vicinity of d = 4 leads to a first order phase transition due to the lack of a stable fixed point [14][15][16] . On the other hand, the low temperature expansion performed around two dimensions on the Non-Linear Sigma (NLσ) model predicts a second order phase transition of the standard O(4)/O(3) universal...

Low-energy properties of two-dimensional quantum triangular antiferromagnets: Nonperturbative renormalization group approach

“…Several theoretical works done by Zumbach, Loison and Schotte, Itakura, and Delamotte et al indicate that the 3D classical triangular HAF show a fluctuation-driven first order phase transition. 41,42,43,44,45,46 On the other hand, the loop expansion calculations carried out by Pelissetto et al and Calabrese et al support the existence of the continuous phase transition in these systems. 47,49 Also, most of experiments seem to be in accordance with the continuous transition.…”

“…In contrast to the quantum case, the three-dimensional (3D) classical stacked Heisenberg model, which is equivalent to the 2D quantum model at T = 0, has been extensively studied by many authors. 8,9,10,14,15,16,17,18,41,42,43,44,45,46,47,48,49 Even for the classical systems, the elucidation of the nature of the phase transition has not yet been completed. Several theoretical works done by Zumbach, Loison and Schotte, Itakura, and Delamotte et al indicate that the 3D classical triangular HAF show a fluctuation-driven first order phase transition.…”

What can be learnt from the nonperturbative renormalization group?

“…A remarkable example is given by the three dimensional frustrated spin systems, either the triangular antiferromagnets or the helimagnets. Here, the results obtained by means of the resummed perturbation series and those obtained from the NPRG are qualitatively different [8][9][10][11][12][13]. The former predicts a second order phase transition whereas the latter very weakly predicts first order transitions.…”

“…when k = 0, Γ k=0 = Γ is the Gibbs free energy of the system since it becomes in this limit the standard Legendre transform of W (R k=0 is identically vanishing); (7) and (8).…”

“…Second, even in cases where five or six loop series are known, resummations do not always lead to converged results. This is the case, for instance, for frustrated spin systems in three dimensions [8][9][10][11][12][13]. Again, contrary to the common belief, the problem is not just to compute accurately critical exponents but to determine the qualitative behavior of the system at the transition: either first or second order phase transition, belonging or not to a given universality class, etc.…”