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arXiv:math-ph/0204011 · Physical review. B, Condensed matter

Ferromagnetic Heisenberg XXZ chain in a pinning field

Pierluigi Contucci, Bruno Nachtergaele, Wolfgang L. Spitzer

Published Aug 30, 2002• 8 citations
arXivHTMLPDF

Abstract

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 ${B}_{c}$ (which depends on the anisotropy of the Hamiltonian and the spin value) with the following properties: for $B<{B}_{c}$ there is a unique ground state with a gap, at the critical value ${B}_{c}$ there are infinitely many (droplet) ground states with gapless excitations, and for $B>{B}_{c}$ 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 $\mathrm{xy}$ 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.

DOI
10.1103/physrevb.66.064429

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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.
Abstract: We consider the ground states of the ferromagnetic XXZ chain with spin up boundary conditions in sectors with a fixed number of down spins. This forces the existence of a droplet of down spins in the system. We find the exact energy and the states that describe these droplets in the limit of an infinite number of down spins. We also prove that the droplet states, which were known to be separated by an energy gap from the ground state, are also separated by an energy gap from the remainder of the spectrum, i.e., they form an isolated band. We also prove the analogous results for finite chains with periodic boundary conditions and for the infinite chain.
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.
For a class of Hamiltonians of XXZ spin chains in a uniform external magnetic field that are small quantum perturbations of an Ising Hamiltonian, it is shown that the spectral gap above the ground-state energy remains strictly positive when the perturbation is turned on, uniformly in the length of the chain. This result is proven for rather generic perturbations of both the ferromagnetic and the antiferromagnetic Ising Hamiltonian. In the antiferromagnetic case, the external magnetic field is required to be small. For a chain of an even number of sites, the two-fold degenerate ground-state energy of the unperturbed antiferromagnetic Hamiltonian may split into two energy levels separated by a very small gap. These results are proven by using a new, quite subtle refinement of a method developed in earlier work and used to iteratively block-diagonalize Hamiltonians of systems confined to ever larger subsets of a lattice by using strictly local unitary conjugations. The new method developed in this paper provides complete control of boundary effects on the low-energy spectrum of perturbed Ising chains uniformly in their length.
We study the open XXZ spin chain in the anti-ferromagnetic regime and for generic longitudinal magnetic fields at the two boundaries. We discuss the ground state via the Bethe ansatz and we show that, for a chain of even length L and in a regime where both boundary magnetic fields are equal and bounded by a critical field, the spectrum is gapped and the ground state is doubly degenerate up to exponentially small corrections in L. We connect this degeneracy to the presence of a boundary root, namely an excitation localized at one of the two boundaries. We compute the local magnetization at the left edge of the chain and we show that, due to the existence of a boundary root, this depends also on the value of the field at the opposite edge, even in the half-infinite chain limit. Moreover we give an exact expression for the large time limit of the spin autocorrelation at the boundary, which we explicitly compute in terms of the form factor between the two quasi-degenerate ground states. This, as we show, turns out to be equal to the contribution of the boundary root to the local magnetization. We finally discuss the case of chains of odd length.
We study the effect of anisotropy of spin-SSS impurities on an antiferromagnetic SU(2) Heisenberg chain. The magnetic impurities are assumed to have an XXZ-type anisotropic exchange with their neighboring sites. First, using density-matrix renormalization group technique, we examine spin-spin correlation function, instability of N\'eel order, and local spin susceptibility in the presence of a single spin-SSS impurity. Based on the results, we find that the types of spin-SSS impurities are classified into two groups: (i) nonmagnetic and S=1S=1S=1 impurities enhance only short-range antiferromagnetic correlation and (ii) S=12S=\frac{1}{2}S=21​ and S>1S>1S>1 impurities can, in contrast, stabilize a long-range N\'eel order in the disordered SU(2) Heisenberg chain. Then, we focus on the case of S=12S=\frac{1}{2}S=21​ impurity as a representative of (ii) and investigate the evolution of some experimentally observable quantities such as magnetization, specific heat, and magnetic susceptibility, as a function of concentration and XXZ anisotropy strength of the impurity. We confirm that the N\'eel order is induced in the bulk spin chain in the presence of finite amount of easy-axis XXZ S=12S=\frac{1}{2}S=21​ impurities. Furthermore, we recover some of the aforementioned features using cluster mean-field theory, which allows us to present results on experimentally accessible quantities at finite temperatures. Interestingly, in the presence of uniform magnetic field, the total magnetization exhibits a pseudogap behavior for low values of applied field. We also discuss the dependence of NMR spectrum on various XXZ impurities and identify that the spin state of Co impurity in SrCu0.99Co0.01O2{\mathrm{SrCu}}_{0.99}{\mathrm{Co}}_{0.01}{\mathrm{O}}_{2}SrCu0.99​Co0.01​O2​ is S=32S=\frac{3}{2}S=23​.
We investigate the spectrum above the kink ground states of the spin J ferromagnetic XXZ chain with Ising anisotropy A. Our main theorem is that there is a non-vanishing gap above all ground states of this model for all values of J. Using a variety of methods, we obtain additional information about the magnitude of this gap, about its behavior for large A, about its overall behavior as a function of A and its dependence on the ground state, about the scaling of the gap and the structure of the low-lying spectrum for large J, and about the existence of isolated eigenvalues in the excitation spectrum.