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arXiv:cond-mat/0309101 · Physical Review B

Nonperturbative renormalization-group approach to frustrated magnets

Bertrand Delamotte, D. Mouhanna, Matthieu Tissier

Published Apr 9, 2004• 204 citations
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Abstract

This article is devoted to the study of the critical properties of classical XY and Heisenberg frustrated magnets in three dimensions. We first analyze the experimental and numerical situations. We show that the unusual behaviors encountered in these systems, typically nonuniversal scaling, are hardly compatible with the hypothesis of a second order phase transition. Moreover, the fact that the scaling laws are significantly violated and that the anomalous dimension is negative in many cases provides strong indications that the transitions in frustrated magnets are most probably of very weak first order. We then review the various perturbative and early nonperturbative approaches used to investigate these systems and argue that none of them provides a completely satisfactory description of the three-dimensional critical behavior. We then recall the principles of the nonperturbative approach---the effective average action method---that we have used to investigate the physics of frustrated magnets and show how it enables to clarify most of the problems encountered in the previous theoretical descriptions of frustrated magnets. First, we get an explanation of the long-standing mismatch between different perturbative approaches which consists in a nonperturbative mechanism of annihilation of fixed points between two and three dimensions. Secondly, we get a coherent picture of the physics of frustrated magnets in agreement with the numerical and experimental results. The central feature that emerges from our approach is the existence of scaling behaviors without fixed or pseudofixed point and that relies on a slowing down of the renormalization group flow in a whole region in the coupling constants space. This phenomenon allow us to explain the occurrence of generic weak first order behaviors and to understand the absence of universality in the critical behavior of frustrated magnets.

DOI
10.1103/physrevb.69.134413

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This article is devoted to the study of the critical properties of classical XY and Heisenberg frustrated magnets in three dimensions. We first analyze the experimental and numerical situations. We show that the unusual behaviors encountered in these systems, typically nonuniversal scaling, are hardly compatible with the hypothesis of a second order phase transition. Moreover, the fact that the scaling laws are significantly violated and that the anomalous dimension is negative in many cases provides strong indications that the transitions in frustrated magnets are most probably of very weak first order. We then review the various perturbative and early nonperturbative approaches used to investigate these systems and argue that none of them provides a completely satisfactory description of the three-dimensional critical behavior. We then recall the principles of the nonperturbative approach---the effective average action method---that we have used to investigate the physics of frustrated magnets and show how it enables to clarify most of the problems encountered in the previous theoretical descriptions of frustrated magnets. First, we get an explanation of the long-standing mismatch between different perturbative approaches which consists in a nonperturbative mechanism of annihilation of fixed points between two and three dimensions. Secondly, we get a coherent picture of the physics of frustrated magnets in agreement with the numerical and experimental results. The central feature that emerges from our approach is the existence of scaling behaviors without fixed or pseudofixed point and that relies on a slowing down of the renormalization group flow in a whole region in the coupling constants space. This phenomenon allow us to explain the occurrence of generic weak first order behaviors and to understand the absence of universality in the critical behavior of frustrated magnets.
XY\mathrm{XY}XY frustrated magnets exhibit an unusual critical behavior: they display scaling laws accompanied by nonuniversal critical exponents and a negative anomalous dimension. This suggests that they undergo weak first-order phase transitions. We show that all perturbative approaches that have been used to investigate XY\mathrm{XY}XY frustrated magnets fail to reproduce these features. Using a nonperturbative approach based on the concept of effective average action, we are able to account for this nonuniversal scaling and to describe qualitatively and, to some extent, quantitatively the physics of these systems.
We show that the critical behavior of two- and three-dimensional frustrated magnets cannot reliably be described from the known five- and six-loop perturbative renormalization-group results. Our conclusions are based on a careful reanalysis of the resummed perturbative series obtained within the zero-momentum massive scheme. In three dimensions, the critical exponents for XY and Heisenberg spins display strong dependences on the parameters of the resummation procedure and on the loop order. This behavior strongly suggests that the fixed points found are in fact spurious. In two dimensions, we find, as in the O(N)O(N)O(N) case, that there is apparent convergence of the critical exponents but toward erroneous values. As a consequence, the interesting question of the description of the crossover/transition induced by Z2{Z}_{2}Z2​ topological defects in two-dimensional frustrated Heisenberg spins remains open.
Abstract We revisit the critical behavior of classical frustrated systems using the nonperturbative renormalization group (NPRG) equation. Our study is performed within the local potential approximation of this equation to which is added the flow of the field renormalization. Our flow equations are functional to avoid possible artifacts coming from the field expansion of the fixed point potential which consists in keeping only a limited number of coupling constants. We explain in detail our numerical implementation, its advantages and the difficulties encountered in the vicinity of d = 2. For N-component spins, the function N c ( d ) separating the regions of first and second order transitions in the (d, N) plane is computed for d between 4 and 2.3. Our results confirm what was previously found with cruder approximations of the NPRG equation and contradict both the fixed dimension perturbative approach and some of the results obtained within the conformal bootstrap approach.
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=2d=2d=2 and d=4d=4d=4. We recover all known perturbative results in a single framework and find the transition to be weakly first order in d=3d=3d=3. We compute effective exponents in good agreement with numerical and experimental data.
We revisit the critical behavior of classical frustrated systems using the nonperturbative renormalization group (NPRG) equation. Our study is performed within the local potential approximation of this equation to which is added the flow of the field renormalization. Our flow equations are functional to avoid possible artifacts coming from field expansions which consists in keeping only a limited number of coupling constants. We present a simple numerical method to follow the fixed point solution of our equations by changing gradually the dimension d and the number N of spin-components. We explain in details the advantage of this method as well as the numerical difficulties we encounter, which become severe close to d = 2. The function N_c(d) separating the regions of first and second order in the (d,N) plane is computed for d between 4 and 2.2. Our results confirm what was previously found within cruder approximation of the NPRG equation and contradict both the fixed dimension perturbative approach and the results obtained within the conformal bootstrap approach.
We show that phase transitions in spin-1 Bose gases and stacked triangular Heisenberg antiferromagnets---an example of frustrated magnets with competing interactions---are described by the same Landau-Ginzburg-Wilson Hamiltonian with $\mathrm{O}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)$ symmetry. In agreement with previous nonperturbative-renormalization-group studies of the three-dimensional $\mathrm{O}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)$ model, we find that the transition from the normal phase to the superfluid ferromagnetic phase in a spin-1 Bose gas is weakly first order and shows pseudoscaling behavior. The (nonuniversal) pseudoscaling exponent $\ensuremath{\nu}$ is fully determined by the scattering lengths a0{a}_{0}a0​ and a2{a}_{2}a2​. We provide estimates of $\ensuremath{\nu}$ in 87Rb,41K^{87}\mathrm{Rb},^{41}\mathrm{K}87Rb,41K, and 7Li^{7}\mathrm{Li}7Li atom gases which can be tested experimentally. We argue that pseudoscaling comes from either a crossover phenomenon due to proximity of the O(6) Wilson-Fisher fixed point (87Rb(^{87}\mathrm{Rb}(87Rb and 41K)^{41}\mathrm{K})41K) or the existence of two unphysical fixed points (with complex coordinates) which slow down the RG flow (7Li)(^{7}\mathrm{Li})(7Li). These unphysical fixed points are a remnant of the chiral and antichiral fixed points that exist in the $\mathrm{O}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)$ model when NNN is larger than ${N}_{c}\ensuremath{\simeq}5.3$ (the transition being then second order and controlled by the chiral fixed point). Finally, we discuss a $\mathrm{O}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)$ lattice model and show that our results, even though we find the transition to be first order, are compatible with Monte Carlo simulations yielding an apparent second-order transition.