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.
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.
Abstract A crucial aspect of the Ising model is its fermionic nature and this chapter is devoted to this property of the model. In the continuum limit, a Dirac equation for neutral Majorana fermions emerges. The details of the derivation are much less important than understanding why it is possible. The chapter emphasizes the simplicity and the exactness of the result, and covers the so-called Wigner-Jordan transformation, which brings the original Hamiltonian to a quadratic form in the creation and annihilation operators of the fermions. It covers the role played by the Bogoliubov transformation and the importance of the order and disorder operators.
Abstract In this chapter we will begin our study of the Ising model on the two-dimensional lattice. In two dimensions the model has a phase transition, with critical exponents that have different values from those obtained in the mean field approximation. For this reason, it provides an important example of critical phenomena. As we will see in great detail in this chapter and in the next, among all exactly solved models of statistical mechanics, the two-dimensional Ising model is not only the one that has been most studied but it is also the model that has given a series of deep mathematical and physical results. Many solutions of the model stand out for the ingenious methods used, such as the theory of determinants, combinatorial approaches, Grassmann variables, or elliptic functions. Many results have deeply influenced the understanding of critical phenomena and have strongly stimulated new fields of research. Ideas that have matured within the study of the two-dimensional Ising model, such as the duality between its highand low-temperature phases, have been readily generalized to other systems of statistical mechanics and have also found important and fundamental applications in other important areas such as, for instance, quantum field theory. Equally fundamental is the discovery that in the vicinity of the critical point, the dynamics of the model can be described from the relativistic Dirac equation for Majorana fermions.
It is shown that a solvable Hamiltonian can be obtained from a series of operators satisfying specific commutation relations. A transformation that diagonalize the Hamiltonian is obtained simultaneously. The two-dimensional Ising model with periodic interactions, the one-dimensional XY model with period 2, the transverse Ising chain, the one-dimensional Kitaev model and the cluster model, and other composite quantum spin chains are diagonalized following this procedure. The Jordan–Wigner transformation, the transformation from the Pauli spin operators to the Majorana fermion used by Shankar and Murthy, and the transformation introduced by Nambu, are special cases of this treatment.
We review the applications of the integral over anticommuting Grassmann variables (nonquantum fermionic fields) to the analytic solutions and the field-theoretical formulations for the 2D Ising models. The 2D Ising model partition function Q is presentable as the fermionic Gaussian integral. The use of the spin-polynomial interpretation of the 2D Ising problem is stressed, in particular. Starting with the spin-polynomial interpretation of the local Boltzmann weights, the Gaussian integral for Q appears in the universal form for a variety of lattices, including the standard rectangular, triangular, and hexagonal lattices, and with the minimal number of fermionic variables (two per site). The analytic solutions for the correspondent 2D Ising models then follow by passing to the momentum space on a lattice. The symmetries and the question on the location of critical point have an interesting interpretation within this spin-polynomial formulation of the problem. From the exact lattice theory we then pass to the continuum-limit field-theoretical interpretation of the 2D Ising models. The continuum theory captures all relevant features of the original models near Tc. The continuum limit corresponds to the low-momentum sector of the exact theory responsible for the critical-point singularities and the large-distance behaviour of correlations. The resulting field theory is the massive two-component Majorana theory, with mass vanishing at Tc. By doubling of fermions in the Majorana representation, we obtain as well the 2D Dirac field theory of charged fermions for 2D Ising models. The differences between particular 2D Ising lattices are merely adsorbed, in the field-theoretical formulation, in the definition of the effective mass.
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 d-dimensional Ising transfer matrix is expressed exactly in terms of Fermi-type operators. In two dimensions this leads to the standard solution of the model. In three or more dimensions the transfer matrix is no longer a quadratic form in the Fermi variables, but rather contains quartic and higher-order terms. However, it can be expressed as a well-defined expansion about a quadratic form in Fermi operators. The zeroth-order transfer-matrix-called herein the "fermion model"---can be exactly diagonalized, and is well approximated in the critical region by ${V}^{(0)}\ensuremath{\approx}{[2sinh(2K)]}^{\frac{N}{2}}\mathrm{exp}\left[\ensuremath{-}N\ensuremath{\int}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}{\left({\ensuremath{\kappa}}^{2}+\ensuremath{\Sigma}{n=1}^{d\ensuremath{-}1}\ensuremath{\Sigma}{m=1}^{d\ensuremath{-}1}{q}_{n}{q}_{m}\right)}^{\frac{1}{2}}t(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})\right]$where N is the number of sites per layer, $\ensuremath{\mu}$ is the inverse correlation length, $t(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ takes values \ifmmode\pm\else\textpm\fi{}1/2, K is the reduced inverse temperature, and the integral extends over a Brillouin zone of the ($d\ensuremath{-}1$)-dimensional layer. The critical point of the model is located by the equation ${x}_{c}^{d}+{x}_{c}^{d\ensuremath{-}1}+{x}_{c}=1$, where ${x}_{c}={e}^{\ensuremath{-}2{K}_{c}}$. The critical exponent $\ensuremath{\nu}$ describing the divergence of the correlation length at Kc is found to take on its two-dimensional value, $\ensuremath{\nu}=1$, in all dimensions d. The possibility of using the fermion model as the basis for an expansion of the Ising critical exponents as a function of $\ensuremath{\delta}(=d\ensuremath{-}2)$ is pointed out.