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Register a new userThe fate of $O(N)$ multi-critical universal behaviour
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The multi-critical fixed points of $O(N)$ symmetric models cease to exist in the $Ntoinfty$ limit, but the mechanism regulating their annihilation still presents several enigmatic aspects. Here, we explore the evolution of high-order multi-critical points in the $(d,N)$ plane and uncover a complex mosaics for their asymptotic behaviour at large $N$. This picture is confirmed by various RG approaches and constitutes a fundamental step towards the full comprehension of critical behaviour in $O(N)$ field theories.
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Using an Environmentally Friendly Renormalization Group we derive an ab initio universal scaling form for the equation of state for the O(N) model, y=f(x), that exhibits all required analyticity properties in the limits $xto 0$, $xtoinfty$ and $xto -1$. Unlike current methodologies based on a phenomenological scaling ansatz the scaling function is derived solely from the underlying Landau-Ginzburg-Wilson Hamiltonian and depends only on the three Wilson functions $gamma_lambda$, $gamma_phi$ and $gamma_{phi^2}$ which exhibit a non-trivial crossover between the Wilson-Fisher fixed point and the strong coupling fixed point associated with the Goldstone modes on the coexistence curve. We give explicit results for N=2, 3 and 4 to one-loop order and compare with known results.
We determine a previously unknown universal quantity, the location of the Yang-Lee edge singularity for the O($N$) theories in a wide range of $N$ and various dimensions. At large $N$, we reproduce the $Ntoinfty$ analytical result on the location of the singularity and, additionally, we obtain the mean-field result for the location in $d=4$ dimensions. In order to capture the nonperturbative physics for arbitrary $N$, $d$ and complex-valued external fields, we use the functional renormalization group approach.
We comment on a recent letter by L. C. de Albuquerque and M. M. Leite (J. Phys. A: Math. Gen. 34 (2001) L327-L332), in which results to second order in $epsilon=4-d+frac{m}{2}$ were presented for the critical exponents $ u_{{mathrm{L}}2}$, $eta_{{mathrm{L}}2}$ and $gamma_{{mathrm{L}}2}$ of d-dimensional systems at m-axial Lifshitz points. We point out that their results are at variance with ours. The discrepancy is due to their incorrect computation of momentum-space integrals. Their speculation that the field-theoretic renormalization group approach, if performed in position space, might give results different from when it is performed in momentum space is refuted.
We demonstrate using direct numerical diagonalization and extrapolation methods that boundary conditions have a profound effect on the bulk properties of a simple $Z(N)$ model for $N ge 3$ for which the model hamiltonian is non-hermitian. For $N=2$ the model reduces to the well known quantum Ising model in a transverse field. For open boundary conditions the $Z(N)$ model is known to be solved exactly in terms of free parafermions. Once the ends of the open chain are connected by considering the model on a ring, the bulk properties, including the ground-state energy per site, are seen to differ dramatically with increasing $N$. Other properties, such as the leading finite-size corrections to the ground-state energy, the mass gap exponent and the specific heat exponent, are also seen to be dependent on the boundary conditions. We speculate that this anomalous bulk behaviour is a topological effect.
We solve analytically the renormalization-group equation for the potential of the O(N)-symmetric scalar theory in the large-N limit and in dimensions 2<d<4, in order to look for nonperturbative fixed points that were found numerically in a recent study. We find new real solutions with singularities in the higher derivatives of the potential at its minimum, and complex solutions with branch cuts along the negative real axis.
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