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arXiv:hep-th/0509149 · Physical Review Letters

Large- n Excitations in the Ferromagnetic Ising Field Theory in a Weak Magnetic Field: Mass Spectrum and Decay Widths

S. B. Rutkevich

Published Dec 15, 2005• 46 citations
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Abstract

In the presence of a small magnetic field h, the elementary excitations in the scaling two-dimensional Ising model are studied perturbatively in in the ferromagnetic phase. For excitations with large numbers n, the mass spectrum is obtained in the first order in h. The decay widths of excitations with energies above the stability threshold are calculated in the leading h3 order.

DOI
10.1103/physrevlett.95.250601

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In the presence of a small magnetic field h, the elementary excitations in the scaling two-dimensional Ising model are studied perturbatively in in the ferromagnetic phase. For excitations with large numbers n, the mass spectrum is obtained in the first order in h. The decay widths of excitations with energies above the stability threshold are calculated in the leading h3 order.
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We study the particle spectrum M n ( h ) in the two-dimensional ferromagnetic Ising field theory in a weak external magnetic field h . According to the Wu and McCoy scenario of the weak confinement, pairs of fermions (domain walls) are coupled into bosonic kink–antikink bound states at small h > 0. Fluctuations with more than two fermions also contribute to the wavefunctions of the compound particles, leading to multi-fermion corrections to their masses M n ( h ) in higher orders in h . We describe a perturbative procedure, which allows us to account for both multi-fermion fluctuations and the long-range confining interaction between fermions, and leads to the formfactor expansions for the renormalized parameters of the model. We obtain integral representations for the third-order multi-fermion correction to the mass M n ( h ), which arises from the regular correction to the kernel of the Bethe–Salpeter equation.
We compute the spectrum and several critical amplitudes of the two dimensional Ising model in a magnetic field with the transfer matrix method. The three lightest masses and their overlaps with the spin and the energy operators are computed on lattices of a width up to L=21. In extracting the continuum results we also take into account the corrections to scaling due to irrelevant operators. In contrast with previous Monte Carlo simulations our final results are in perfect agreement with the predictions of S-matrix and conformal field theory. We also obtain the amplitudes of some of the subleading corrections, for which no S-matrix prediction has yet been obtained.
The main subject of this thesis is the Ising field theory, the field theory describing the scaling limit of the two-dimensional Ising model near its critical point. We study the Ising field theory in low- and high-temperature regimes.In the low-temperature regime we address questions related to the mass spectrum of the bound states that exist in the theory. The overall goal is to study analytic properties of the masses taken as functions of the scaling variable in the theory. At this stage, we are primarily interested in the asymptotic series in the coupling parameter. We describe methods used for developing expansions in the coupling parameter when this parameter is sufficiently small. This gives rise to the low-energy and semiclassical series. Methods developed are then applied to a different model, a model that has many things in common with the Ising field theory, the two-dimensional quantum chromodynamics with infinite number of colors, also known as the ‘t Hooft model.In the second part of the thesis we turn to the Ising field theory in the high- temperature regime. Here we study the problem of scattering in a weak magnetic field. We compute the leading perturbative correction to the scattering amplitude for the 2 -> 2 process. This is done first by means of standard methods based on the direct intermediate-state decomposition and dispersion relations. Then we compute the large energy asymptotic of the amplitude for this process directly using a known relation between the Ising field theory at zero magnetic field and classical sinh-Gordon equation. We obtain consistent results that show the logarithmic growth of the amplitude with energy at this order in magnetic field. Going beyond the leading order we argue that at large energies for a sufficiently weak magnetic field the amplitude exhibits a power-like decay.