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arXiv:cond-mat/0203533

Comment on ``Bicritical and Tetracritical Phenomena and Scaling Properties of the SO(5) Theory,'' [X. Hu, Phys. Rev. Lett. 87, 057004 (2001)]

Pasquale Calabrese, Andrea Pelissetto C, Ettore Vicari

Published Mar 26, 2002• 4 citations
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Abstract

We show that, in the most general $N$-component theory with symmetry O(n_1)+O(n_2), N=n_1+n_2\geq 3, the O(N)-symmetric fixed point has (at least) three unstable directions: the temperature, the quadratic anisotropy, and the spin-4 quartic perturbation. This implies that in the SO(5) theory of high-$T_c$ superconductivity an additional tuning is required to obtain an enlarged SO(5) symmetry. This is in contrast with the recent numerical results reported by Hu, Phys. Rev. Lett. 87, 057004 (2001). \

DOI
10.48550/arxiv.cond-mat/0203533

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Comment on “Bicritical and Tetracritical Phenomena and Scaling Properties of the SO(5) Theory”

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The multicritical point at which both a 3-component and a 2-component order parameters order simultaneously in 3 dimensions is shown to have the critical behavior of the decoupled fixed point, with separate n=3 and n=2 behavior. This contradicts both the extrapolation of the epsilon-expansion at leading order, which yields the biconical point, and recent Monte Carlo simulations, which gave isotropic SO(5) behavior. Thus, this tetracritical point carries no information on the relevance of the so-called SO(5) theory of high-T superconductivity.

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Lian-Ao Wu, Mike Guidry, Yang Sun +1 more

An SU(4) model of high-temperature superconductivity and antiferromagnetism has recently been proposed. The SO(5) group employed by Zhang [Science 275, 1089 (1997)] is embedded in this SU(4) as a subgroup, suggesting a connection between our SU(4) model and the Zhang SO(5) model. In order to understand the relationship between the the two models, we have used generalized coherent states to analyze the nature of the SO(5) subgroup. By constructing coherent-state energy surfaces, we demonstrate explicitly that the $\mathrm{SU}(4)\ensuremath{\supset}\mathrm{SO}(5)$ symmetry can be interpreted as a critical dynamical symmetry interpolating between superconducting and antiferromagnetic phases, and that this critical dynamical symmetry has many similarities to critical dynamical symmetries identified previously in other fields of physics. More generally, we demonstrate with this example that the mathematical techniques associated with generalized coherent states may have powerful applications in condensed-matter physics because they provide a clear connection between microscopic many-body theories and their broken-symmetry approximate solutions. In addition, these methods may be interpreted as defining the most general Bogoliubov transformation subject to a Lie group symmetry constraint, thus providing a mathematical connection between algebraic formulations and the language of quasiparticle theory. Finally, we suggest that the identification of the SO(5) symmetry as a critical dynamical symmetry implies deep algebraic connections between high-temperature superconductors and seemingly unrelated phenomena in other field of physics.

arXiv:cond-mat/0208329•Physical review. B, Condensed matter•Jan 17, 2003•24 citations
We show that, in the most general NNN-component theory with symmetry O(n_1)+O(n_2), N=n_1+n_2\geq 3, the O(N)-symmetric fixed point has (at least) three unstable directions: the temperature, the quadratic anisotropy, and the spin-4 quartic perturbation. This implies that in the SO(5) theory of high-TcT_cTc​ superconductivity an additional tuning is required to obtain an enlarged SO(5) symmetry. This is in contrast with the recent numerical results reported by Hu, Phys. Rev. Lett. 87, 057004 (2001). \
The multicritical point at which both a 3-component and a 2-component order parameters order simultaneously in 3 dimensions is shown to have the critical behavior of the decoupled fixed point, with separate n=3 and n=2 behavior. This contradicts both the extrapolation of the epsilon-expansion at leading order, which yields the biconical point, and recent Monte Carlo simulations, which gave isotropic SO(5) behavior. Thus, this tetracritical point carries no information on the relevance of the so-called SO(5) theory of high-T superconductivity.
We study the nature of the multicritical point in the three-dimensional $\mathrm{O}(3)\ensuremath{\bigoplus}\mathrm{O}(2)$ symmetric Landau-Ginzburg-Wilson theory, which describes the competition of two order parameters that are O(3) and O(2) symmetric, respectively. This study is relevant for SO(5) theory of high-Tc{T}_{c}Tc​ superconductors, which predicts the existence of a multicritical point in the temperature-doping phase diagram, where the antiferromagnetic and superconducting transition lines meet. We investigate whether $\mathrm{O}(3)\ensuremath{\bigoplus}\mathrm{O}(2)$ symmetry gets effectively enlarged to O(5) approaching the multicritical point. For this purpose, we study the stability of the O(5) fixed point. By means of a Monte Carlo simulation, we show that the O(5) fixed point is unstable with respect to the spin-4 quartic perturbation with the crossover exponent ${\ensuremath{\phi}}_{4,4}=0.180(15)$, in substantial agreement with recent field-theoretical results. This estimate is much larger than the one-loop ϵϵϵ-expansion estimate ${\ensuremath{\phi}}_{4,4}=1∕26$, which has often been used in the literature to discuss the multicritical behavior within SO(5) theory. Therefore, no symmetry enlargement is generically expected at the multicritical transition. We also perform a five-loop field-theoretical analysis of the renormalization-group flow. It shows that bicritical systems are not in the attraction domain of the stable decoupled fixed point. Thus, in these systems---high-Tc{T}_{c}Tc​ cuprates should belong to this class---the multicritical point corresponds to a first-order transition.
By Monte Carlo simulations on a 3D SO(5) rotator model it is shown that, in contrast to the epsilon expansions of the renormalization group, the bicritical point is stable to biquadratic perturbations of AF-SC (antiferromagnetism-superconductivity) repulsions, which are produced by quantum fluctuations originated from the Gutzwiller projection. Therefore, the present work completes the link from a physical projected SO(5) starting point to an asymptotic SO(5) symmetry point. The tetracritical point is stable for attractive AF-SC interactions. Critical exponents and ratios are evaluated by scaling analysis. Bicritical and tetracritical scaling functions are derived for the first time. Suggestions on experiments are given.
We study the multicritical behavior arising from the competition of two distinct types of ordering characterized by O(n)\mathrm{O}(n)O(n) symmetries. For this purpose, we consider the renormalization-group flow for the most general $\mathrm{O}{(n}_{1})\ensuremath{\bigoplus}\mathrm{O}{(n}_{2})$-symmetric Landau-Ginzburg-Wilson Hamiltonian involving two fields ${\ensuremath{\varphi}}_{1}$ and ${\ensuremath{\varphi}}_{2}$ with n1{n}_{1}n1​ and n2{n}_{2}n2​ components, respectively. In particular, we determine in which cases, approaching the multicritical point, one may observe the asymptotic enlargement of the symmetry to O(N)\mathrm{O}(N)O(N) with N=n1+n2.{N=n}_{1}{+n}_{2}.N=n1​+n2​. By performing a five-loop $\ensuremath{\epsilon}$-expansion computation we determine the fixed points and their stability. It turns out that for N=n1+n2> 3{N=n}_{1}{+n}_{2}>~3N=n1​+n2​> 3 the O(N)\mathrm{O}(N)O(N)-symmetric fixed point is unstable. For N=3,N=3,N=3, the multicritical behavior is described by the biconal fixed point with critical exponents that are very close to the Heisenberg ones. For N> 4N>~4N> 4 and any n1,n2{n}_{1}{,n}_{2}n1​,n2​ the critical behavior is controlled by the tetracritical decoupled fixed point. We discuss the relevance of these results for some physically interesting systems, in particular for anisotropic antiferromagnets in the presence of a magnetic field and for high-Tc{T}_{c}Tc​ superconductors. Concerning the SO(5) theory of superconductivity, we show that the bicritical O(5) fixed point is unstable with a significant crossover exponent ${\ensuremath{\varphi}}_{4,4}\ensuremath{\approx}0.15;$ this implies that the O(5) symmetry is not effectively realized at the point where the antiferromagnetic and superconducting transition lines meet. The multicritical behavior is either governed by the tetracritical decoupled fixed point or is of first-order type if the system is outside its attraction domain.
Thirty years after the Liu-Fisher paper on the {\bf bicritical} and {\bf tetracritical} points in quantum lattice gases, these multicritical points continue to appear in a variety of new physical contexts. This paper reviews some recent multicritical phase diagrams, which involve e. g. high-TcT_cTc​ superconductivity and various magnetic phases which may (or may not) coexist with it. One recent example concerns the SO(5) theory, which combines the 3-component antiferromagnetic and the 2-component superconducting order parameters. There, the competition between the isotropic, biconical and decoupled fixed points yields bicritical or tetracritical points. Recalling old results on the subject, it is shown that the {\bf decoupled} fixed point is stable, implying a {\bf tetracritical} point, contrary to recent claims, which are critically discussed. Other examples, concerning e. g. the superconducting versus charge and spin density wave phases are also discussed briefly. In all cases, extensions of old results can be used to correct new claims.
An SU(4) model of high-temperature superconductivity and antiferromagnetism has recently been proposed. The SO(5) group employed by Zhang [Science 275, 1089 (1997)] is embedded in this SU(4) as a subgroup, suggesting a connection between our SU(4) model and the Zhang SO(5) model. In order to understand the relationship between the the two models, we have used generalized coherent states to analyze the nature of the SO(5) subgroup. By constructing coherent-state energy surfaces, we demonstrate explicitly that the $\mathrm{SU}(4)\ensuremath{\supset}\mathrm{SO}(5)$ symmetry can be interpreted as a critical dynamical symmetry interpolating between superconducting and antiferromagnetic phases, and that this critical dynamical symmetry has many similarities to critical dynamical symmetries identified previously in other fields of physics. More generally, we demonstrate with this example that the mathematical techniques associated with generalized coherent states may have powerful applications in condensed-matter physics because they provide a clear connection between microscopic many-body theories and their broken-symmetry approximate solutions. In addition, these methods may be interpreted as defining the most general Bogoliubov transformation subject to a Lie group symmetry constraint, thus providing a mathematical connection between algebraic formulations and the language of quasiparticle theory. Finally, we suggest that the identification of the SO(5) symmetry as a critical dynamical symmetry implies deep algebraic connections between high-temperature superconductors and seemingly unrelated phenomena in other field of physics.