In the present thesis we perform a step towards understanding the effec-
tiveness of Hamiltonian perturbation theory in quantum dynamics, which is made possible by the Hamiltonian framing of quantum mechanics. To such a purpose, we focus our attention on quantum spin chains, namely on the isotropic Heisenberg ferromagnet, which is well known to be equivalent, via the Jordan-Wigner transformation, to a system of interacting fermions, and is also known to be exactly integrable through the Bethe ansatz. The interaction term of the Hamiltonian, of order four in fermionic creation and annihilation operators, is treated as a perturbation in the regime of small
excitations: small number of reversed spins or, equivalently, small number of
fermions per site. The calculation of the first order normal form Hamiltonian
amounts to erasing all the nonresonant interaction terms, and allows to draw
interesting conclusions on the dynamics of the system, e.g. to determine ap-
proximately conserved quantities. Moreover, we get a good correspondence
between our perturbative energy spectrum and the exact one computed via
the Bethe ansatz, the approximation being of course the more accurate the
smaller is the number of fermions per site. The present method, here tested
in an exactly solvable case, can be applied to any nonintegrable system of
weakly interacting fermions (and/or bosons).
Integrable quantum spin chains are exactly solvable quantum mechanical models of N quantum spins, of which the Heisenberg model solved by Bethe is the prototype. In the antiferromagnetic regime, such spin chains can be regarded as lattice versions of corresponding integrable relativistic quantum field theories. For integrable spin chains, quantities of physical interest (spectrum, S matrix, etc.) can be calculated exactly by direct means, starting from the microscopic Hamiltonian, while for the corresponding field theories, such exact information has been primarily obtained by indirect means, such as the “ bootstrap ” approach and semiclassical approximations. We have been focused basically on the Heisenberg XXX and XXZ chains and their generalizations. Basic aims of this work were the study of the symmetries of these models and the computation, by means of Bethe ansatz equations, of the bulk and boundary S matrices for the closed and open chains respectively...
Introduction to Integrable Many-Body Systems II This is the second part of a three-volume introductory course about integrable systems of interacting bodies. The models of interest are quantum spin chains with nearest-neighbor interactions between spin operators, in particular Heisenberg spin-1/2 models. The Ising model in a transverse field, expressible as a quadratic fermion form by using the Jordan-Wigner transformation, is the subject of Sect. 12. The derivation of the coordinate Bethe ansatz for the XXZ Heisenberg chain and the determination of its absolute ground state in various regions of the anisotropy parameter are presented in Sect. 13. The magnetic properties of the ground state are explained in Sect. 14. Sect. 15 concerns excited states and the zero-temperature thermodynamics of the XXZ model. The thermodynamics of the XXZ Heisenberg chain is derived on the basis of the string hypothesis in Sect. 16; the thermodynamic Bethe ansatz equations are analyzed in high-temperature and low-temperature limits. An alternative derivation of the thermodynamics without using strings, leading to a non-linear integral equation determining the free energy, is the subject of Sect. 17. A nontrivial application of the Quantum Inverse Scattering method to the fully anisotropic XYZ Heisenberg chain is described in Sect. 18. Sect. 19 deals with integrable cases of isotropic spin chains with an arbitrary spin.
With the help of the Jordan–Wigner transformation the spin-12 XY chains can be reformulated in terms of noninteracting spinless fermions and, as a result, many statistical-mechanics calculations can be performed rigorously, i.e. without making any simplifying approximations. We are interested in dynamic properties of such chains (time-dependent spin correlation functions, dynamic structure factors, dynamic susceptibilities) which are of great importance for interpretation of experimental data. We have worked out a number of dynamic quantities explicitly as well as have performed a general analysis of the two-fermion continua which are relevant for different dynamic quantities.
Since a long-time, the quantum integrable systems have remained an area where modern mathematical methods have given an access to interesting results in the study of physical systems. The exact computations, both numerical and asymptotic, of the correlation function is one of the most important subject of the theory of the quantum integrable models. In this context an approach based on the calculation of form factors has been proved to be a more effective one. In this thesis, we develop a new method based on the algebraic Bethe ansatz for the computation of the form-factors in thermodynamic limit. It is both applied to and described in the context of isotropic XXX Heisenberg chain, which is one of the examples of an interesting case of critical models where the Fermi-zone is non-compact. In a particular case of two-spinon form-factors, we obtain an exact result in a closed-form which matches the previous result obtained from an approach based on
q-vertex operator algebra. This method is then generalised to form-factors in higher spinon sectors where we find a reduced determinant representation for the form-factors, in which a higher-level structure for the form-factors is revealed.
Quantum spin chains arise naturally from perturbative large- N field theories and matrix models. The Hamiltonian of such a model is a long-range deformation of nearest-neighbour type interactions. Here, we study the most general long-range integrable spin chain with spins transforming in the fundamental representation of . We derive the Hamiltonian and the corresponding asymptotic Bethe ansatz at the leading four perturbative orders with several free parameters. Furthermore, we propose Bethe equations for all orders and identify the moduli of the integrable system. We finally apply our results to plane-wave matrix theory and show that the Hamiltonian in a closed sector is not of this form and therefore not integrable beyond the first perturbative order. This also implies that the complete model is not integrable.
A general system composed of many weakly interacting bosons and/or fermions is considered. The object is to develop a procedure for renormalizing the single-particle creation operators, so as to remove the interactions to successively higher orders of perturbation. The basic idea is that creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ must satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where
H is the Hamiltonian and ${\ensuremath{\omega}}_{i}$ are the particle energies. For the many-weakly-interacting-particle system, the zeroth-order boson and fermion creation operators satisfy these equations to zeroth order. It is shown that if the creation operators are renormalized so as to satisfy the Hamiltonian commutator equations to order
m, and also to satisfy the appropriate boson commutators and fermion anticommutators to order
m, then the problem is solved to order
m. In particular, the vectors formed by operating on the ground state with the renormalized creation operators, according to the usual boson and fermion occupation-number representation, are eigenvectors of
H and are orthonormal, all to order
m. A procedure is otained for finding the (
m+1)-order contributions to the particle creation energies, in terms of the
m-order operators. Explicit first-order calculations of these general results are provided for a system of bosons and a system of fermions, and these first-order results are shown to include similar results of Rayleigh-Schr\"odinger perturbation theory, the random-phase approximation, and the method of thermodynamic Green's functions. The problem of anharmonic lattice dynamics is studied in detail, and a method of undetermined coefficients is used to renormalize the phonon creation operators to first order. The phonon energies are calculated to second order, and this calculation shows that the interactions between renormalized phonons cannot be removed in second order. Statistical averages of the phonon energies give the energy shifts and lifetimes which have been calculated previously by various propagator techniques. In addition, the renormalized energies are used to calculate the temperature-dependent part of the Helmholtz free energy correct to second order. As a final example, electron-phonon interactions in a normal metal are studied. The electron and phonon creation operators are renormalized to first order, the particle energies are calculated to second order, and statistical averages of the particle energies recover the usual thermodynamic Green's-function results for energy shifts and lifetimes. These examples show the simplicity by which the renormalization procedure obtains a great amount of detailed information about the single-particle nature of a many-particle system.