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Ground State Of The Heisenberg XXZ Chain With Two PinningFields

Matthew Dominique Mitchell

Published Jan 1, 2019• 0 citations
OpenAlex

Abstract

We begin by providing a brief introduction to quantum spin systems including the de nitions and outlining the framework in which we'll be working for the rest of the thesis. We de ne and provide examples of F-functions and reproduce the proof of the Lieb-Robinson bound. [1] We then consider the Heisenberg XXZ spin-1/2 chain model and prove the form of the ground state space with kink and anti- kink boundary conditions. With the addition of a single pinning field, we demonstrate how this selects a single state in the ground state space to be the new unique ground state.

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We begin by providing a brief introduction to quantum spin systems including the de nitions and outlining the framework in which we'll be working for the rest of the thesis. We de ne and provide examples of F-functions and reproduce the proof of the Lieb-Robinson bound. [1] We then consider the Heisenberg XXZ spin-1/2 chain model and prove the form of the ground state space with kink and anti- kink boundary conditions. With the addition of a single pinning field, we demonstrate how this selects a single state in the ground state space to be the new unique ground state.
We investigate the effect of a magnetic field supported at a single lattice site on the low-energy spectrum of the ferromagnetic Heisenberg XXZ chain. Such fields, caused by impurities, can modify the low-energy spectrum significantly by pinning certain excitations, such as kink and droplet states. We distinguish between different boundary conditions (or sectors), the direction and also the strength of the magnetic field. E.g., with a magnetic field in the z direction applied at the origin and ++ boundary conditions, there is a critical field strength Bc{B}_{c}Bc​ (which depends on the anisotropy of the Hamiltonian and the spin value) with the following properties: for B<BcB<{B}_{c}B<Bc​ there is a unique ground state with a gap, at the critical value Bc{B}_{c}Bc​ there are infinitely many (droplet) ground states with gapless excitations, and for B>BcB>{B}_{c}B>Bc​ there is again a unique ground state but now belonging to the continuous spectrum. In contrast, any magnetic field with a nonvanishing component in the xy\mathrm{xy}xy plane yields a unique ground state, which, depending on the boundary conditions, is either an (anti)kink, or an (anti)droplet state. For such fields, i.e., not aligned with the z axis, excitations always have a gap and we obtain a rigorous lower bound for that gap.
This thesis is divided into 4 chapters. The first chapter gives a brief discussion on the need for the study of Quantum spin systems, and an introduction to Heisenberg Model. In the second chapter, we start the discussion with Quantum entanglement and then go on to show the importance of anti ferromagnetic ground state from the view of entanglement. In the third chapter, we discuss the ways to obtain the ground states of Heisenberg model, both ferromagnet and antiferromagnet. Then we discuss the effect of magnetic field on the antiferromagnetic ground states, for a good understanding of magnetic properties in Chapter(4). In the fourth and the final chapter, we discuss how Bonner Fisher model uses the Bethe ansatz to obtain all the properties of finite Heisenberg chains. Then we analyze the magnetic properties, like Magnetic moment and Susceptibility, of Heisenberg antiferromagnets for finite as well as infinite chains.
We show that the well-known translation invariant ground states and the recently discovered kink and antikink ground states are the complete set of pure infinite-volume ground states (in the sense of local stability) of the spin-5 ferromagnetic XXZ chains with Hamiltonian H = -EJS^S^ 4-Si 2) S% + ASi^S^l for all A > 1, and all 5 € |N.For the isotropic model (A = 1) we show that all ground states are translation invariant.For the proof of these statements we propose a strategy for demonstrating completeness of the list of the pure infinite-volume ground states of a quantum many-body system, of which the present results for the XXX and XXZ chains can be seen as an example.The result for A > 1 can also be proved by an easy extension to general S of the method used in [T.Matsui, Lett.Math.Phys.37 (1996) 397] for the spin-1/2 ferromagnetic XXZ chain with A > 1.However, our proof is e-print archive: