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arXiv:hep-th/9610161 · Journal of Physics A Mathematical and General

Functional integral approach to multipoint correlators in two-dimensional critical systems

Victoria Fernández, Carlos Naón

Published Mar 21, 1997• 2 citations
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Abstract

We extend a previously developed technique for computing spin - spin critical correlators in the two-dimensional (2D) Ising model, to the case of multiple correlations. This enables us to derive Kadanoff - Ceva's formula in a simple and elegant way. We also exploit a doubling procedure in order to evaluate the critical exponent of the polarization operator in the Baxter model. Thus we provide a rigorous proof of the relation between different exponents, in the path-integral framework.

DOI
10.1088/0305-4470/30/6/011

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We extend a previously developed technique for computing spin - spin critical correlators in the two-dimensional (2D) Ising model, to the case of multiple correlations. This enables us to derive Kadanoff - Ceva's formula in a simple and elegant way. We also exploit a doubling procedure in order to evaluate the critical exponent of the polarization operator in the Baxter model. Thus we provide a rigorous proof of the relation between different exponents, in the path-integral framework.
We construct a relationship between the Baxter model in two dimensions and the Luttinger model in one, and use it to calculate critical exponents for the Baxter model from appropriate Luttinger-model correlation functions. An important part of this work involves the generalization of the Jordan-Wigner transformation to provide a representation for continuum spin operators. With this generalization, we are also able to calculate spin correlation functions for a continuum generalization of the spin-\textonehalf{} Heisenberg-Ising chain. We discuss the difference between the continuum and discrete lattice models, and with the help of a new scaling law, use previous results for the Baxter model to calculate new exponents for the Baxter and Heisenberg-Ising model on a lattice.
The relation between critical exponents and the amplitude of the correlation length divergence at the critical point of two-dimensional systems as a function of finite size is investigated and generalised in two ways. Correlations more general than those of order-order type are included. A form appropriate for anisotropic systems is proposed. The authors present (a) exact results for the Ising and Gaussian models and (b) numerical results for the symmetric eight-vertex (Baxter), continuous q-state Potts, and continuous N-component cubic models.
We consider the critical spin-spin correlation function of the Ashkin-Teller and Baxter models. By using path-integral techniques in the continuum description of these models in terms of fermion fields, we show that the correlation decays with distance with the same critical exponent as the Ising model. The procedure is straightforwardly extended to take into account the presence of a line defect. Thus we find that in these altered models the critical index of the magnetic correlation on the defect coincides with the one of the defective two-dimensional Ising or Bariev model.
The nearest-neighbor-interaction ferromagnetic Ashkin-Teller model is investigated on a square lattice through a powerful computational method for dealing with correlation functions in magnetic systems. This technique, which is based on damage-spreading numerical simulations, makes use of exact relations involving special kinds of damage and correlation functions, as well as the corresponding order parameters of the model. The computation of correlation functions, which represents usually a hard task in standard Monte Carlo simulations, due to large fluctuations, turns out to be much simpler within the present approach. We concentrate our analysis along the Baxter line, well known for its continuously varying critical exponents; seven different points along this line are investigated. The critical exponents associated with correlation functions along the Baxter line are successfully evaluated, by means of numerical methods, within damage-spreading simulations. The efficiency of this method is confirmed through precise estimates of the critical exponents associated with the order parameters (magnetization and polarization), as well as with their corresponding correlation functions, in spite of the small lattice sizes considered.
Recently it has been found that for several two-dimensional classical models the amplitude of the correlation length of a finite system at a critical point is universal and related to the correlation-function exponent $\ensuremath{\eta}$. Here we propose that for quantum systems the ratios of two such amplitudes are universal and are related to the ratios of the corresponding exponents. This proposition is confirmed numerically for the transverse Ising model for spin S=12,1,32,and2S=\frac{1}{2},1,\frac{3}{2}, \mathrm{and} 2S=21​,1,23​,and2. It is used to calculate $\ensuremath{\eta}$ in a simplified way.
In this paper we review the critical Ising model by using the properties of conformal invariance. We use the known mapping of the Ising model to a theory of Majorana fermions and recognize that the self-duality property appears as a double degeneracy of the periodic ground state. The Ising model is doubled obtaining an Ashkin-Teller model (two Ising models coupled with a four-spin coupling) at the decoupling point. Using the bosonization technique this model is mapped onto the Gaussian model at a particular value of the temperature (K=1/\ensuremath{\pi}). This allows us to give the expressions for products of order, disorder, and energy operators of both Ising models in terms of operators in the Gaussian model. We compute several correlation functions of order, disorder, energy and spinor operators and show that they reproduce the operator product expansions predicted by conformal invariance. We explicitly discuss the physics by which the correlation functions obtained from the Gaussian model (a theory with conformal anomaly c=1) reproduce those of the Ising model (c=1/2). This provides a proof of Kadanoff and Brown's equivalences and conjectures. We provide a simple prescription to compute all n-point correlation functions at criticality, including mixed correlations of order, disorder, energy, and spinor operators. A bosonized form of the continuum limit of the transfer matrix for the critical Ising model is constructed. It is nonlocal and contains both spin-wave and vortex operators of the Gaussian model.