Using the density matrix renormalization group technique, we study the ground state phase diagram and other low-energy properties of an isotropic antiferromagnetic spin-half chain with both dimerization and frustration, i.e., an alternation δ of the nearest neighbor exchanges and a next-nearest-neighbor exchange J 2 . For δ = 0, the system is gapless for J 2 < J 2c and has a gap for J 2 > J 2c where J 2c is about 0.241.For J 2 = J 2c , the gap above the ground state grows as δ to the power 0.667 ± 0.001. In the J 2 − δ plane, there is a disorder line 2J 2 + δ = 1. To the left of this line, the peak in the static structure factor S(q) is at q max = π (Neel phase), while to the right of the line, q max decreases from π to π/2 as J 2 is increased to large values (spiral phase). For δ = 1, the system is equivalent to two coupled chains as on a ladder and it is gapped for all values of the interchain coupling.

We show how Majorana end modes can be generated in a one-dimensional system by varying some of the parameters in the Hamiltonian periodically in time. The specific model we consider is a chain containing spinless electrons with a nearest-neighbor hopping amplitude, a p-wave superconducting term and a chemical potential; this is equivalent to a spin-1/2 chain with anisotropic XY couplings between nearest neighbors and a magnetic field applied in the z-direction. We show that varying the chemical potential (or magnetic field) periodically in time can produce Majorana modes at the ends of a long chain. We discuss two kinds of periodic driving, periodic delta-function kicks and a simple harmonic variation with time. We discuss some distinctive features of the end modes such as the inverse participation ratio of their wave functions and their Floquet eigenvalues which are always equal to +/- 1 for time-reversal symmetric systems. For the case of periodic delta-function kicks, we use the effective Hamiltonian of a system with periodic boundary conditions to define two topological invariants. The first invariant is a well-known winding number while the second invariant has not appeared in the literature before. The second invariant is more powerful in that it always correctly predicts the numbers of end modes with Floquet eigenvalues equal to +1 and -1, while the first invariant does not. We find that the number of end modes can become very large as the driving frequency decreases. We show that periodic delta-function kicks in the hopping and superconducting terms can also produce end modes. Finally, we study the effect of electron-phonon interactions (which are relevant at finite temperatures) and a random noise in the chemical potential on the Majorana modes.Comment: 15 pages, 11 figures; added more numerical and analytical results about second topological invariant, and a discussion of effects of electron-phonon interactions and noise on Majorana end mode

We present a unified study of the effect of periodic, quasiperiodic, and disordered potentials on topological phases that are characterized by Majorana end modes in one-dimensional p-wave superconducting systems. We define a topological invariant derived from the equations of motion for Majorana modes and, as our first application, employ it to characterize the phase diagram for simple periodic structures. Our general result is a relation between the topological invariant and the normal state localization length. This link allows us to leverage the considerable literature on localization physics and obtain the topological phase diagrams and their salient features for quasiperiodic and disordered systems for the entire region of parameter space.

We report spin wave and DMRG studies of the ground and low-lying excited states of uniform and dimerized alternating spin chains. The DMRG procedure is also employed to obtain low-temperature thermodynamic properties of the system. The ground state of a 2N spin system with spin-1 and spin-1 2 alternating from site to site and interacting via an antiferromagnetic exchange is found to be ferrimagnetic with total spin s G = N/2 from both DMRG and spin wave analysis. Both the studies also show that there is a gapless excitation to a state with spin s G − 1 and a gapped excitation to a state with spin s G + 1. Surprisingly, the correlation length in the ground state is found to be very small from both the studies for this gapless system. For this very reason, we show that the ground state can be described by a variational ansatz of the product type. DMRG analysis shows that the chain is susceptible to a conditional spin-Peierls' instability. The DMRG studies of magnetization, magnetic susceptibility (χ) and specific heat show strong magnetic-field dependence. The product χT shows a minimum as a function of temperature(T ) at low-magnetic fields and the minimum vanishes at high-magnetic fields. This lowfield behaviour is in agreement with earlier experimental observations. The specific heat shows a maximum as a function of temperature and the height of the maximum increases sharply at high-magnetic fields. It is hoped that these studies will motivate experimental studies at high-magnetic fields.