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Physical Review Letters

Renormalization-Group Analysis of Bicritical and Tetracritical Points

David R. Nelson, J. Michael Kosterlitz, Michael Ellis Fisher

Published Sep 30, 1974• 185 citations
Publisher

Abstract

Recently developed renormalization-group techniques are summarized and exploited to yield a renormalization-group analysis of bicritical and tetracritical points (which arise in antiferromagnets and boson systems). For $n\ensuremath{\lesssim}3$ an isotropic or Heisenberg fixed point dominates and gives bicritical behavior; but for $n\ensuremath{\gtrsim}4$ a new "biconical" fixed point with irrational $\ensuremath{\epsilon}$-expansion coefficients appears. This describes a tetracritical point and may be relevant to displacive phase transitions.

DOI
10.1103/physrevlett.33.813

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arXiv:2203.01168•Low Temperature Physics•May 24, 2022•4 citations
Recently developed renormalization-group techniques are summarized and exploited to yield a renormalization-group analysis of bicritical and tetracritical points (which arise in antiferromagnets and boson systems). For n\ensuremath≲3n\ensuremath{\lesssim}3n\ensuremath≲3 an isotropic or Heisenberg fixed point dominates and gives bicritical behavior; but for n\ensuremath≳4n\ensuremath{\gtrsim}4n\ensuremath≳4 a new "biconical" fixed point with irrational \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-expansion coefficients appears. This describes a tetracritical point and may be relevant to displacive phase transitions.
We discuss the static and dynamic multicritical behavior of three-dimensional systems of O(n∥)⊕O(n⊥)O(n_\|)\oplus O(n_\perp)O(n∥​)⊕O(n⊥​) symmetry as it is explained by the field theoretical renormalization group method. Whereas the static renormalization group functions are currently know within high order expansions, we show that an account of two loop contributions refined by an appropriate resummation technique gives an accurate quantitative description of the multicritical behavior. One of the essential features of the static multicritical behavior obtained already in two loop order for the interesting case of an antiferromagnet in a magnetic field (n∥=1n_\|=1n∥​=1, n⊥=2n_\perp=2n⊥​=2) are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points leading to very small transient exponents. We further pursue an analysis of dynamical multicritical behavior choosing different forms of critical dynamics and calculating asymptotic and effective dynamical exponents within the minimal subtraction scheme.
We calculate the static critical behavior of systems of O(n_||)(plus sign in circle)O(n_perpendicular) symmetry by the renormalization group method within the minimal subtraction scheme in two-loop order. Summation methods lead to fixed points describing multicritical behavior. Their stability border lines in the space of the order parameter components n_|| and n_perpendicular and spatial dimension d are calculated. The essential features obtained already in two-loop order for the interesting case of an antiferromagnet in a magnetic field ( n_|| =1, n_perpendicular =2 ) are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points, leading to very small transient exponents. We are also able to calculate the flow of static couplings, which allows us to consider the attraction region. Depending on the nonuniversal background parameters, the existence of different multicritical behavior (bicritical or tetracritical) is possible, including a triple point.
Renormalization-group theory predicts that the XXZ antiferromagnet in a magnetic field along the easy ZZZ axis has asymptotically either a tetracritical phase diagram or a triple point in the field-temperature plane. Neither experiments nor Monte Carlo simulations procure such phase diagrams. Instead, they find a bicritical phase diagram. Here, this discrepancy is resolved: After generalizing a ubiquitous condition identifying the tetracritical point, we employ different renormalization-group recursion relations near the isotropic fixed point, exploiting group-theoretical considerations and using accurate exponents at three dimensions. These show that the results from experiments and simulations can only be understood if their trajectories flow towards the fluctuation-driven first-order transition (and the associated triple point), but reach this limit only for prohibitively large system sizes or correlation lengths. In the crossover region one expects a bicritical phase diagram, as indeed is observed. A similar scenario may explain puzzling discrepancies between simulations and renormalization-group predictions for a variety of other phase diagrams with competing order parameters.
This article concludes a series of papers [Folk, Holovatch, and Moser, Phys. Rev. E 78, 041124 (2008); 78, 041125 (2008); 79, 031109 (2009)] where the tools of the field theoretical renormalization group were employed to explain and quantitatively describe different types of static and dynamic behavior in the vicinity of multicritical points. Here we give the complete two-loop calculation and analysis of the dynamic renormalization-group flow equations at the multicritical point in anisotropic antiferromagnets in an external magnetic field. We find that the time scales of the order parameters characterizing the parallel and perpendicular ordering with respect to the external field scale in the same way. This holds independent whether the Heisenberg fixed point or the biconical fixed point in statics is the stable one. The nonasymptotic analysis of the dynamic flow equations shows that due to cancellation effects the critical behavior is described, in distances from the critical point accessible to experiments, by the critical behavior qualitatively found in one-loop order. Although one may conclude from the effective dynamic exponents (taking almost their one-loop values) that weak scaling for the order parameter components is valid, the flow of the time-scale ratios is quite different, and they do not reach their asymptotic values.
A scaling theory is introduced for bicritical points, such as antiferromagnetic spinflop points (with analogies to the upper $\ensuremath{\lambda}$ point in 4He^{4}\mathrm{He}4He), where two distinct critical lines meet. Experimentally testable predictions follow from renormalization-group calculations which indicate that the bicritical exponents should be Heisenberg like for systems with $n\ensuremath{\le}3$ components; the crossover exponent $\ensuremath{\varphi}(\ensuremath{\simeq}1.25 \mathrm{for} n=3)$ is directly observable. For n>3n>3n>3 an intermediate ("supersolid") low-temperature phase may appear, the bicritical point then becoming tetracritical.
The critical behavior of many physical systems involves two competing n1- and n2-component order-parameters, S1 and S2, respectively, with n = n1 + n2. Varying an external control parameter g, one encounters ordering of S1 below a critical (second-order) line for g < 0 and of S2 below another critical line for g > 0. These two ordered phases are separated by a first-order line, which meets the above critical lines at a bicritical point, or by an intermediate (mixed) phase, bounded by two critical lines, which meet the above critical lines at a tetracritical point. For n = 1 + 2 = 3, the critical behavior around the (bi- or tetra-) multicritical point either belongs to the universality class of a non-rotationally invariant (cubic or biconical) fixed point, or it has a fluctuation driven first-order transition. These asymptotic behaviors arise only very close to the transitions. We present accurate renormalization-group flow trajectories yielding the effective crossover exponents near multicriticality.