Integrable quantum spin chains

Date of Award




Degree Name

Doctor of Philosophy (Ph.D.)



First Committee Member

Rafael I. Nepomechie, Committee Chair

Second Committee Member

L. Mezincescu, Committee Member


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. We give a simplified derivation of the boundary S-matrix in the XXX and XXZ chain by exploiting the existence of the "odd" sector (i.e., N odd), in which the number of excitations is odd. We also demonstrate the factorization of multiparticle scattering for the XXX chain. We compute the three-particle S-matrix, and we show that it is factorizable into a product of two-particle S matrices.We generalize our computations for SU( N ) integrable spin chain. For the closed SU( N ) chain we describe the multiparticle states, and we explicitly determine the SU( N ) quantum numbers of the states. We show that the model has particle-like excitations in the fundamental representations of SU( N ), and we directly compute the complete two-particle S matrices.We consider the open A&parl0;1&parr0;N-1 integrable chain with diagonal boundary magnetic fields. The boundary magnetic fields break the quantum group symmetry U q(SU( N )) to lower symmetry Uq (SU( N -- 1)) x Uq (SU(1)) x U(1). Moreover, we find a new "duality" symmetry which maps 1↔N-1 . With the help of these residual symmetries of the model, we compute the corresponding boundary S matrices, which describe scattering from the ends of the chain. This is the first direct calculation of boundary S matrices for a model whose symmetry algebra has rank greater than one.We formulate the notion of parity and charge conjugation for the periodic XXZ spin chain. We use these discrete symmetries to classify low-lying states (critical regime), and we compute the S matrix elements corresponding to these states.Finally, we formulate a systematic Bethe-Ansatz approach for computing bound-state ("breather") S matrices for integrable quantum spin chains. We use this approach to calculate the breather boundary S matrix for the open XXZ spin chain with diagonal boundary fields. We also compute the soliton boundary S matrix in the critical regime.


Physics, Elementary Particles and High Energy

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