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Physical Review Letters

Dependence of Critical Indices on a Parameter

Robert B. Griffiths

Published Jun 29, 1970• 332 citations
Publisher

Abstract

A number of lattice models of phase transitions and critical points can be related to one another by variation of linear parameters appearing in the Hamiltonian. It is suggested that the critical indices as functions of these parameters remain constant except possibly at points where the nature of the associated first-order phase transition is itself altered by varying a parameter. Several models exhibiting critical phenomena are consistent with this proposal.

DOI
10.1103/physrevlett.24.1479

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A number of lattice models of phase transitions and critical points can be related to one another by variation of linear parameters appearing in the Hamiltonian. It is suggested that the critical indices as functions of these parameters remain constant except possibly at points where the nature of the associated first-order phase transition is itself altered by varying a parameter. Several models exhibiting critical phenomena are consistent with this proposal.
In the context of a phase transition, the critical singularity is characterized by a set of numbers, the critical exponents. It is widely believed that for transitions in systems with pair interactions on Bravais lattices (as opposed to hierarchical lattices), the critical exponents are determined solely by parameters d and n, representing the dimensionality of space and of the order parameter, respectively, and by \ensuremath{\sigma}, the interaction range. In this paper we consider a certain generalization of the Lifshitz singularity in the n-vector model in which a new parameter \ensuremath{\lambda}, describing a type of anisotropy of the interactions, is introduced. This is described by a model Hamiltonian in k space in which the leading term is of the form [${\mathrm{\ensuremath{\Sigma}}}_{\mathit{i}=1}^{\mathit{d}}$(ki2{\mathit{k}}_{\mathit{i}}^{2}ki2​${)}^{\mathrm{\ensuremath{\sigma}}/(2\ensuremath{\lambda})}$]\ensuremath{\lambda}. It should be noted that, as for the usual Lifshitz case, what would otherwise be the dominant k2{\mathit{k}}^{2}k2 term is supposed to have already been cancelled. For this model the parameters d, n, and \ensuremath{\sigma} are not sufficient to determine the value of the critical exponents to O(1/n), although they are sufficient for the first order in the \ensuremath{\epsilon} expansion.Calculations exact to O(1/n) show that, for a given set d, n, and \ensuremath{\sigma}, the exponents vary continuously with the parameter \ensuremath{\lambda} and include as a special case (for \ensuremath{\lambda}=1) the critical exponents for an anisotropic Lifshitz point. The critical exponents for this Lifshitz point differ markedly in value from those for the usually studied isotropic Lifshitz point, which is also a special case (for \ensuremath{\lambda}=\ensuremath{\sigma}/2) of our model. For \ensuremath{\sigma}=2, however, the two cases corresponding to \ensuremath{\lambda}=1 and \ensuremath{\sigma}/2 become degenerate. We provide another explicit example of the dependence of the critical exponents on \ensuremath{\lambda} by performing, in addition to the 1/n expansion, an expansion in \ensuremath{\sigma}-2. This helps to shed some light on the approach to the degeneracy at \ensuremath{\sigma}=2. The analysis of the Feynman integrals arising in the O(1/n) calculation provides interesting examples of the applications of hypergeometric functions; these integrals have themselves led to new results (published separately) in the theory of hypergeometric functions.
In this paper we briefly review our finding about the effects of boundary conditions and lattice shapes on scaling functions in critical phenomena. We present new examples to show that different boundary conditions and lattice shapes may give quite different scaling functions for lattice phase transition models, but they give consistent critical point, critical exponents, and thermodynamic order parameter from renormalization group calculations. We also review our recent results about the universal scaling func-tions for bond and site percolations on planar lattices with different boundary conditions and lattice shapes. PACS. 05.5O.+q- Lattice theory and statistics; Ising problems. PACS. 64.60.Fr- Equilibrium properties near critical points, critical exponents. PACS. 75.10.Hk- Classical spin models. I.
We consider a chain of localized spins, coupled to phonons. Recently this problem has been solved exactly for a ``basic model,'' a family of spin-phonon Hamiltonians HBM{\mathit{H}}_{\mathrm{BM}}HBM​ characterized by one parameter (coupling constant K), and a zero-T first-order phase transition from the magnetic (ferro or antiferro) state at low couplings to the nonmagnetic state with a phonon density wave at high couplings was found. Here we probe the general case, constructing an effective Hamiltonian H for low-energy degrees of freedom by means of regular expansion in deviations \ensuremath{\delta}H=H-HBM{\mathit{H}}_{\mathrm{BM}}HBM​ of the general Hamiltonian H from that of the basic model. In linear approximation in \ensuremath{\delta}H the problem appears to be exactly solvable as well, due to an infinite number of conservation laws. If K is far enough from the critical value Kc{\mathit{K}}_{\mathit{c}}Kc​, then the character of the basic model solution is not altered. In the vicinity of Kc{\mathit{K}}_{\mathit{c}}Kc​ the magnetic state is dramatically reconstructed: Here the ground state is a gapless magnetic quantum liquid, consisting of mobile singlet spin-phonon complexes and unbound spins. The fraction of singlets increases gradually upon approaching Kc{\mathit{K}}_{\mathit{c}}Kc​, and the magnetic order parameter gradually vanishes. Thus we have here a partial screening of spins by phonons without formation of a phonon density wave. The latter appears only at K=Kc{\mathit{K}}_{\mathit{c}}Kc​ in the first-order phase transition. Corrections, quadratic in \ensuremath{\delta}H, destroy the integrability of the system, but outside a narrow critical region around Kc{\mathit{K}}_{\mathit{c}}Kc​ they only lead to an opening of a small gap in the spectrum of the quantum liquid. The behavior of the system within the critical region is an open question. Most likely the continuous magnetic phase transition at K=Kc{\mathit{K}}_{\mathit{c}}Kc​ becomes a first-order one, but close to second order. The relevance of our results for three-dimensional systems and possible applications to compounds with anomalously weak magnetism are briefly discussed. \textcopyright{} 1996 The American Physical Society.
Thermodynamic perturbation theory is applied to systems that undergo a phase transition. It is shown that certain conditions are necessary in order that the application of the perturbation does not change the qualitative features of the phase transition, e.g., the critical indices. The method is then applied to the two-dimensional-square Ising lattice with nearest-and next-nearest-neighbor interactions. The shift in the critical temperature is determined to two orders in the peturbation parameter, viz., the ratio of the strength of the next-nearest-neighbor force to the strength of the nearest-neighbor force. A similar calculation is also made for a two-dimensional-lattice potential of Kac and Helfand.
There is a number of contradictory findings with regard to whether the theory describing easy-plane quantum antiferromagnets undergoes a second-order phase transition. The traditional Landau-Ginzburg-Wilson approach suggests a first-order phase transition, as there are two different competing order parameters. On the other hand, it is known that the theory has the property of self-duality which has been connected to the existence of a deconfined quantum critical point (DQCP). The latter regime suggests that order parameters are not the elementary building blocks of the theory, but rather consist of fractionalized particles that are confined in both phases of the transition and only appear-deconfine-at the critical point. Nevertheless, many numerical Monte Carlo simulations disagree with the claim of a DQCP in the system, indicating instead a first-order phase transition. Here we establish from exact lattice duality transformations and renormalization group analysis that the easy-plane CP^{1} antiferromagnet does feature a DQCP. We uncover the criticality starting from a regime analogous to the zero temperature limit of a certain classical statistical mechanics system which we therefore dub frozen. At criticality our bosonic theory is dual to a fermionic one with two massless Dirac fermions, which thus undergoes a second-order phase transition as well.