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arXiv:1210.4037 · Physical Review Letters

O(N) -Universality Classes and the Mermin-Wagner Theorem

Alessandro Codello, Giulio D’Odorico

Published Apr 5, 2013• 77 citations
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Abstract

We study how universality classes of O(N)-symmetric models depend continuously on the dimension d and the number of field components N. We observe, from a renormalization group perspective, how the implications of the Mermin-Wagner-Hohenberg theorem set in as we gradually deform theory space towards d = 2. For a fractal dimension in the range 2 < d < 3, we find, for any N ≥ 1, a finite family of multicritical effective potentials of increasing order. Apart from the N = 1 case, these disappear in d = 2 consistently with the Mermin-Wagner-Hohenberg theorem. Finally, we study O(N = 0)-universality classes and find an infinite family of these in two dimensions.

DOI
10.1103/physrevlett.110.141601

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We study how universality classes of O(N)-symmetric models depend continuously on the dimension d and the number of field components N. We observe, from a renormalization group perspective, how the implications of the Mermin-Wagner-Hohenberg theorem set in as we gradually deform theory space towards d = 2. For a fractal dimension in the range 2 < d < 3, we find, for any N ≥ 1, a finite family of multicritical effective potentials of increasing order. Apart from the N = 1 case, these disappear in d = 2 consistently with the Mermin-Wagner-Hohenberg theorem. Finally, we study O(N = 0)-universality classes and find an infinite family of these in two dimensions.
We compute critical exponents of O(N)O(N)O(N) models in fractional dimensions between d=2d=2d=2 and 4, and for continuous values of the number of field components NNN, in this way completing the RG classification of universality classes for these models. These curves represent nonperturbative approximation to the exact results, they respect all the qualitative features expected from such quantities conciliating previously known perturbative results in three dimensions with exact results in two dimensions and giving a strong indication of what could be the exact behavior of such curves. We also report critical exponents for some multicritical universality classes in the cases $N\ensuremath{\ge}2$ and N=0N=0N=0. Finally, in the large-NNN limit our critical exponents correctly approach those of the spherical model, allowing us to set $N\ensuremath{\sim}100$ as the threshold for the quantitative validity of leading order large-NNN estimates.
We study the Blume-Capel universality class in $d=\frac{10}{3}\ensuremath{-}\ensuremath{\epsilon}$ dimensions. The renormalization group flow is extracted by looking at poles in fractional dimension of three loop diagrams using MS‾\overline{\mathrm{MS}}MS. The theory is the only nontrivial universality class which admits an expansion to three dimensions with $\ensuremath{\epsilon}=\frac{1}{3}<1$. We compute the relevant scaling exponents and estimate some of the operator product expansion coefficients to the leading order. Our findings agree with and complement conformal field theory results. Finally we discuss a family of nonunitary multicritical models which includes the Lee-Yang and Blume-Capel classes as special cases.
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Dynamical renormalization group analysis of O(n)O(n)O(n) model in steady shear flow
We study the critical behavior of the O(n) O ( n ) model under steady shear flow using a dynamical renormalization group (RG) method. Incorporating the strong anisotropy in scaling ansatz, which has been neglected in earlier RG analyses, we identify a new stable Gaussian fixed point. This fixed point reproduces the anisotropic scaling of static and dynamical critical exponents for both non-conserved (Model A) and conserved (Model B) order parameters. Notably, the upper critical dimensions are d_{\text{up}} = 2 d up = 2 for the non-conserved order parameter (Model A) and d_{\text{up}} = 0 d up = 0 for the conserved order parameter (Model B), implying that the mean-field critical exponents are observed even in both d=2 d = 2 and 3 3 dimensions. Furthermore, the scaling exponent of the order parameter is negative for all dimensions d ≥ 2 d ≥ 2 , indicating that shear flow stabilizes the long-range order associated with continuous symmetry breaking even in d = 2 d = 2 . In other words, the lower critical dimensions are d_{\rm low} < 2 for both types of order parameters. This contrasts with equilibrium systems, where the Hohenberg—Mermin—Wagner theorem prohibits continuous symmetry breaking in d = 2 d = 2 .
Spontaneous symmetry breaking can persist at all temperatures in certain biconical O(N)×Z_{2} vector models when the underlying field theories are ultraviolet complete. So far, the existence of such theories has been established in fractional dimensions for local but nonunitary models or in 2+1 dimensions but for nonlocal models. Here, we study local models at zero and finite temperature directly in 2+1 dimensions employing functional methods. At zero temperature, we establish that our approach describes the quantum critical behavior with good accuracy for all N≥2. We then exhibit the mechanism of discrete symmetry breaking from O(N)×Z_{2}→O(N) for increasing temperature near the biconical critical point when N is finite but large. We calculate the corresponding finite-temperature phase diagram and further show that the Hohenberg-Mermin-Wagner theorem is fully respected within this approach, i.e., symmetry breaking only occurs in the Z_{2} sector. Finally, we determine the critical N above which this phenomenon can be observed to be N_{c}≈15.
Abstract The Mermin-Wagner theorem states that spontaneous continuous symmetry breaking is prohibited in systems with short-range interactions at spatial dimension D ≤ 2. For long-range interactions with a power-law form (1/ r α ), the theorem further forbids ferromagnetic or antiferromagnetic order at finite temperature when α ≥ 2 D . However, the situation for α ∈ (2, 4) at D = 2 is not covered by the theorem. To address this, we conduct large-scale quantum Monte Carlo simulations and field theoretical analysis. Our findings show spontaneous breaking of S U (2) symmetry in the ferromagnetic Heisenberg model with 1/ r α -form long-range interactions at D = 2. We determine critical exponents through finite-size analysis for α < 3 (above the upper critical dimension with Gaussian fixed point) and 3 ≤ α < 4 (below the upper critical dimension with non-Gaussian fixed point). These results reveal new critical behaviors in 2D long-range Heisenberg models, encouraging further experimental studies of quantum materials with long-range interactions beyond the Mermin-Wagner theorem’s scope.