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Nuclear Physics B

The order/disorder quantum field operators associated with the two-dimensional Ising model in the continuum limit

Bert Schroer, Trong Tuong Truong

Published Nov 1, 1978• 110 citations
Publisher
DOI
10.1016/0550-3213(78)90499-6

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The quantum field theoretical formulation of Kadanoff’s “correlations along a line” in the critical two-dimensional Ising model is given in terms of p-products of Wightman fields. It provides a quadratic factorization, in the sense of operator-valued distributions, of Abelian chiral vertex operators into non-Abelian exchange fields. This basic result implies p-polynomial relations between fields from various conformal models in two dimensions; notably the two-dimensional local fields with chiral SU(2) symmetry at level 2 are shown to share the nontrivial peculiarities of the Ising model order and disorder fields. These examples have interesting implications for the properties of p-products in general.

arXiv:hep-th/9705033•Journal of Mathematical Physics•Jun 1, 1998•2 citations
Abstract In this chapter we will study the continuum limit formulation of the two-dimensional Ising model, starting from the hamiltonian limit of its transfer matrix. We will first derive the quantum hamiltonian of the model and then we will study its most important properties, such as the duality transformation. This symmetry involves the order and disorder operators and we will clarify their physical interpretation. Afterwards, we will see how to diagonalize the quantum hamiltonian by means of particular fermionic fields. The operator mapping between the order/disorder operators and the fermionic fields is realized by the so-called Wigner–Jordan transformation: this brings the original hamiltonian to a quadratic form in the creation and annihilation operators of the fermions. The determination of the spectrum is then obtained by a Bogoliubov transformation, a technique extremely useful also in other contexts, such as superconductivity phenomena. In the limit in which the lattice spacing goes to zero, the Ising model becomes a theory of free Majorana fermions. They satisfy a relativistic dispersion relation and their mass is a direct measurement of the displacement of the temperature from the critical value.
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.
The quantum field theoretical formulation of Kadanoff’s “correlations along a line” in the critical two-dimensional Ising model is given in terms of p-products of Wightman fields. It provides a quadratic factorization, in the sense of operator-valued distributions, of Abelian chiral vertex operators into non-Abelian exchange fields. This basic result implies p-polynomial relations between fields from various conformal models in two dimensions; notably the two-dimensional local fields with chiral SU(2) symmetry at level 2 are shown to share the nontrivial peculiarities of the Ising model order and disorder fields. These examples have interesting implications for the properties of p-products in general.