Ferromagnetic XXZ Chain Research Articles

Intrinsic Localized Modes in Quantum Ferromagnetic XXZ Chains in an Oblique Magnetic Field

A semiclassical study of intrinsic localized spin-wave modes in a one-dimensional quantum ferromagnetic XXZ chain in an oblique magnetic field is presented in this paper. We quantize the model Hamiltonian by introducing the Dyson-Maleev transformation, and adopt the coherent state representation as the basic representation of the system. By means of the method of multiple scales combined with a quasidiscreteness approximation, the equation of motion for the coherent-state amplitude can be reduced to the standard nonlinear Schrodinger equation. It is found that, at the center of the Brillouin zone, when θ θc a dark intrinsic localized spin-wave resonance mode can occur above the bottom of the magnon frequency band. In other words, the switch between the bright and dark intrinsic localized spin-wave modes can be controlled via varying the angle of the magnetic field. This result has potential applications in quantum information storage. In addition, we find that, at the boundary of the Brillouin zone, the system can only produce a dark intrinsic localized spin-wave mode, whose eigenfrequency is above the upper of the magnon frequency band.

Universal adiabatic quantum computation via the space-time circuit-to-Hamiltonian construction.

We show how to perform universal adiabatic quantum computation using a Hamiltonian which describes a set of particles with local interactions on a two-dimensional grid. A single parameter in the Hamiltonian is adiabatically changed as a function of time to simulate the quantum circuit. We bound the eigenvalue gap above the unique ground state by mapping our model onto the ferromagnetic XXZ chain with kink boundary conditions; the gap of this spin chain was computed exactly by Koma and Nachtergaele using its q-deformed version of SU(2) symmetry. We also discuss a related time-independent Hamiltonian which was shown by Janzing to be capable of universal computation. We observe that in the limit of large system size, the time evolution is equivalent to the exactly solvable quantum walk on Young's lattice.

Implementing quantum gates using the ferromagnetic spin-J XXZ chain with kink boundary conditions

We demonstrate an implementation scheme for constructing quantum gates using unitary evolutions of the one-dimensional spin-J ferromagnetic XXZ chain. We present numerical results based on simulations of the chain using the time-dependent density matrix renormalization group method and techniques from optimal control theory. Using only a few control parameters, we find that it is possible to implement one- and two-qubit gates on a system of spin- XXZ chains, such as Not, Hadamard, Pi-8, Phase and C-Not, with fidelity levels exceeding 99%.

Instability of Interfaces in the Antiferromagnetic XXZ Chain at Zero Temperature

For the antiferromagnetic, highly anisotropic XZ and XXZ quantum spin chains, we impose periodic boundary conditions on chains with an odd number of sites to force an interface (or kink) into the chain. We prove that the energy of the interface depends on the momentum of the state. This shows that at zero temperature the interface in such chains is not stable. This is in contrast to the ferromagnetic XXZ chain for which the existence of localized interface ground states has been proven for any amount of anisotropy in the Ising-like regime.

Improved Bounds on the Spectral Gap Above Frustration-Free Ground States of Quantum Spin Chains

Nachtergaele obtained explicit lower bounds for the spectral gap above many frustration free quantum spin chains by using the ‘martingale method’. We present simple improvements to his main bounds which allow one to obtain a sharp lower bound for the spectral gap above the spin-1/2 ferromagnetic XXZ chain. As an illustration of the method, we also calculate a lower bound for the spectral gap of the AKLT model, which is about 1/3 the size of the expected gap.

Ferromagnetic HeisenbergXXZchain in a pinning field

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.

Droplet States in the XXZ Heisenberg Chain

We consider the ground states of the ferromagnetic XXZ chain with spin up boundary conditions. The ground state of this model, restricted to a sector with a fixed number of down spins, describes a droplet of down spins in an environment of up spins. We find the exact energy and the states that describe these droplets in the limit of an infinite number of down spins. We prove that there is a gap in the spectrum above the droplet states. As the XXZ Hamiltonian has a gap above the fully magnetized ground states as well, this means that the droplet states (for sufficiently large droplets) form an isolated band. The width of this band tends to zero in the limit of infinitely large droplets. We also prove the analogous results for finite chains with periodic boundary conditions and for the infinite chain.

The spectral gap for the ferromagnetic spin-J XXZ chain

We investigate the spectrum above the kink ground states of the spin J ferromagnetic XXZ chain with Ising anisotropy �. 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 be- havior for large �, about its overall behavior as a function ofand 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. By combining in- formation obtained by perturbation theory, numerical, and asymptotic analysis we arrive at a number of interesting conjectures. The proof of the main theorem, as well as some of the numerical results, rely on a comparison result with a Solid-on-Solid (SOS) approximation. This SOS model itself raises interesting questions in combinatorics, and we believe it will prove useful in the study of interfaces in the XXZ model in higher dimensions.

The complete set of ground states of the ferromagnetic XXZ 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-S ferromagnetic XXZ chains with Hamiltonian H=-sum_x [ S^1_x S^1_{x+1} + S^2_x S^2_{x+1} + Delta S^3_x S^3_{x+1} ], for all Delta >1, and all S=1/2,1,3/2,.... For the isotropic model (Delta =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 Delta>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 $\Delta>1$. However, our proof is different and does not rely on the existence of a spectral gap. In particular, it also works to prove absence of non-translationally invariant ground states for the isotropic chains (Delta=1), which have a gapless excitation spectrum. Our results show that, while any small amount of the anisotropy is enough to stabilize the domain walls against the quantum fluctuations, no boundary condition exists that would stabilize a domain wall in the isotropic model (Delta=1).

The Spectral Gap of the Ferromagnetic XXZ-Chain

We prove that the spectral gap of the spin-\(\frac{1}{2}\) ferromagnetic XXZ-chain with HamiltonianH=−Σ_x S^{(1)}_xS^{(1)}_{x+1}+S^{(2)}_xS^{(2)}_{x+1}+\Delta S^{(3)}_xS^{(3)}_{x+1}, is given by Δ-1 for all Δ≥1. This is the gap in the spectrum of the infinite chainin any of its ground states, the translation invariant ones as well asthe kink ground states, which contain an interface between an up and a down region.In particular, this shows that the lowest magnon energy is not affected by the presence of a domain wall. This surprising fact is a consequence of the SU q (2)quantum group symmetry of the model.