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Physical Review Letters

Spin Flop, Supersolids, and Bicritical and Tetracritical Points

Michael Ellis Fisher, David R. Nelson

Published Jun 17, 1974• 359 citations
Publisher

Abstract

A scaling theory is introduced for bicritical points, such as antiferromagnetic spinflop points (with analogies to the upper $\ensuremath{\lambda}$ point in $^{4}\mathrm{He}$), 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 $n>3$ an intermediate ("supersolid") low-temperature phase may appear, the bicritical point then becoming tetracritical.

DOI
10.1103/physrevlett.32.1350

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A scaling theory is introduced for bicritical points, such as antiferromagnetic spinflop points (with analogies to the upper \ensuremathλ\ensuremath{\lambda}\ensuremathλ 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≤3n\ensuremath{\le}3n\ensuremath≤3 components; the crossover exponent \ensuremathφ(\ensuremath≃1.25forn=3)\ensuremath{\varphi}(\ensuremath{\simeq}1.25 \mathrm{for} n=3)\ensuremathφ(\ensuremath≃1.25forn=3) is directly observable. For n>3n>3n>3 an intermediate ("supersolid") low-temperature phase may appear, the bicritical point then becoming tetracritical.
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.
Crossover scaling functions for a two-component spin system with quadratic anisotropy ggg and cubic anisotropy $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$ are calculated to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ and in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$, near the transitions within the ordered phases below the multicritical point (T=Tm,g=0T={T}_{m},g=0T=Tm​,g=0). Emphasis is placed on the mechanism by which the system is driven to bicritical or tetracritical behavior by the dangerous irrelevant variable $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$. Explicit expressions and graphs for the various thermodynamic scaling functions are presented.
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.
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.
We study the critical dynamics of stochastic models appropriate for describing bicritical and tetracritical points in anisotropic antiferromagnetic systems. The dynamic exponents and the transient exponents are calculated by renormalized field theory up to two-loop order. In the presence of reversible mode-coupling terms, two-loop contributions establish bicritical dynamic scaling in the restricted sense and invalidate recent predictions based on mode-coupling arguments. In the case of an n-component relaxational model total dynamic scaling is found to O (ε2) both at the bicritical (n≲3) and the tetracritical (4≲n<11) points.
We calculate by renormalized field theory, up to two-loop order, the bicritical dynamic exponents of stochastic models appropriate for describing spin-flop bicritical points. In the presence of reversible mode-coupling terms, two-loop contributions establish bicritical dynamic scaling in the restricted sense and invalidate recent predictions based on mode-coupling arguments. In case of a purely relaxational model, total bicritical dynamic scaling is found to second order in $\ensuremath{\epsilon}=4\ensuremath{-}d$.