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arXiv:0901.1571 · Journal of Physics A Mathematical and Theoretical

Formfactor perturbation expansions and confinement in the Ising field theory

S. B. Rutkevich

Published Jul 14, 2009• 18 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8113/42/30/304025

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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.
In the ferromagnetic phase of the q-state Potts model, switching on an external magnetic field in- duces confinement of the domain wall excitations. For the Ising model (q = 2) the spectrum consists of kink-antikink states which are the analogues of mesonic states in QCD, while for q = 3, depending on the sign of the field, the spectrum may also contain three-kink bound states which are the analogues of the baryons. In recent years the resulting “hadron” spectrum was described using several different approaches, such as quantum mechanics in the confining linear potential, WKB methods and also the Bethe-Salpeter equation. Here we compare the available predictions to numerical results from renor- malization group improved truncated conformal space approach (RG-TCSA). While mesonic states in the Ising model have already been considered in a different truncated Hamiltonian approach, this is the first time that a precision numerical study is performed for the 3-state Potts model. We find that the semiclassical approach provides a very accurate description for the mesonic spectrum in all the parameter regime for weak magnetic field, while the low-energy expansion from the Bethe-Salpeter equation is only valid for very weak fields where it gives a slight improvement over the semiclassical results. In addition, we confirm the validity of the recent predictions for the baryon spectrum obtained from solving the quantum mechanical three-body problem.
In the ferromagnetic phase of the q-state Potts model, switching on an external magnetic field induces confinement of the domain wall excitations. For the Ising model (q = 2) the spectrum consists of kink-antikink states which are the analogues of mesonic states in QCD, while for q = 3, depending on the sign of the field, the spectrum may also contain three-kink bound states which are the analogues of the baryons. In recent years the resulting “hadron” spectrum was described using several different approaches, such as quantum mechanics in the confining linear potential, WKB methods and also the Bethe-Salpeter equation. Here we compare the available predictions to numerical results from renormalization group improved truncated conformal space approach (RG-TCSA). While mesonic states in the Ising model have already been considered in a different truncated Hamiltonian approach, this is the first time that a precision numerical study is performed for the 3-state Potts model. We find that the semiclassical approach provides a very accurate description for the mesonic spectrum in all the parameter regime for weak magnetic field, while the low-energy expansion from the Bethe-Salpeter equation is only valid for very weak fields where it gives a slight improvement over the semiclassical results. In addition, we confirm the validity of the recent predictions for the baryon spectrum obtained from solving the quantum mechanical three-body problem.
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.
Weakly coupled Ising chains provide a condensed-matter realization of confinement. In these systems, kinks and antikinks bind into mesons due to an attractive interaction potential that increases linearly with the distance between the particles. While single mesons have been directly observed in experiments, the role of the multiparticle continuum and bound states of mesons in the excitation spectrum is far less clear. Using time-dependent density-matrix renormalization group methods, we study the dynamical structure factors of one- and two-spin operators in a transverse-field two-leg Ising ladder in the ferromagnetic phase. The propagation of time-dependent correlations and the two-spin excitation spectrum reveal the existence of interchain bound states, which are absent in the one-spin dynamical structure factor. We also identify two-meson bound states that appear at higher energies, above the thresholds of several two-meson continua.
A bstract When two transverse-field Ising chains (TFICs) with magnetic order are coupled, the original free excitations become confined, giving rise to meson-like bound states. In this work, we study such bound states systematically. The mesons are characterized by their fermion number parity and chain-exchanging properties, which lead to distinct sets of mesonic states. The meson masses are determined by solving the Bethe-Salpter equation. An interesting observation is the additional degeneracy in the chain-exchanging odd sectors. Beyond the two particle approximation, we exploit the truncated free fermionic space approach to calculate the spectrum numerically. Corrections to the meson masses are obtained, and the degeneracy is further confirmed. The characterization and degeneracy can be connected to the situation when each chain is tuned to be quantum critical, where the system is described by the Isingh2 {\textrm{Ising}}_h^2 Isingh2​ Ising h 2 integrable model, a sine-Gordon theory with ℤ 2 orbifold. Here we establish a clear correspondence between the particles in the bosonized form and their fermionic counterparts. Near this point, the stability of these particles is analyzed using the form factor perturbation scheme, where four particles are always present. Additionally, we calculate the evolution of the dominant dynamical structure factor for local spin operators, providing further insight into the low-energy excitations and their role in the system’s behavior. The two-particle confinement framework as well as the parity classifications may inspire the study for other coupled bi-partite systems.