We study Majorana chain with the shortest possible interaction term and in the presence of hopping alternation. When formulated in terms of spins the model corresponds to the transverse field Ising model with nearest-neighbor transverse and next-nearest-neighbor longitudinal repulsion. The phase diagram obtained with extensive DMRG simulations is very rich and contains six phases. Four gapped phases include paramagnetic, period-2 with broken translation symmetry, \mathbb{Z}_2 ℤ 2 with broken parity symmetry and the period-2- \mathbb{Z}_2 ℤ 2 phase with both symmetries broken. In addition there are two floating phases: gapless and critical Luttinger liquid with incommensurate correlations, and with an additional spontaneously broken \mathbb{Z}_2 ℤ 2 symmetry in one of them. By analyzing an extended phase diagram we demonstrate that, in contrast with a common belief, the Luttinger liquid phase along the self-dual critical line terminates at a weaker interaction strength than the end point of the Ising critical line that we find to be in the tri-critical Ising universality class. We also show that none of these two points is a Lifshitz point terminating the incommensurability. In addition, we analyzed topological properties through Majorana zero modes emergent in the two topological phases, with and without incommensurability. In the weak interaction regime, a self-consistent mean-field treatment provides a remarkable accuracy for the description of the spectral pairing and the parity switches induced by the interaction.
Majorana zero modes (MZM-s) occurring at the edges of a one-dimensional (1D),
p-wave, spinless superconductor, in the absence of fluctuations of the phase of the superconducting order parameter, are quintessential examples of topologically protected zero-energy modes occurring at the edges of 1D symmetry-protected topological phases. In this work, we numerically investigate the fate of the topological phase in the presence of phase fluctuations using the density matrix renormalization group (DMRG) technique. To that end, we consider a one-dimensional array of MZM-s on mesoscopic superconducting islands at zero temperature. Cooper-pair and MZM-assisted single-electron tunneling, together with finite charging energy of the mesoscopic islands, give rise to a rich phase diagram of this model. We show that the system can be in either a Mott-insulating phase, a Luttinger liquid (LL) phase of Cooper pairs, or a second gapless phase. In contrast to the LL of Cooper pairs, this second phase is characterized by nonlocal string correlation functions which decay algebraically due to gapless charge-
e excitations. The three phases are separated from each other by phase transitions of either Kosterlitz-Thouless or Ising type. Using a Jordan-Wigner transformation, we map the system to a generalized Bose-Hubbard model with two types of hopping and use DMRG to analyze the different phases and the phase transitions.
We study Kitaev model in one-dimension with open boundary condition by using exact analytic methods for non-interacting system at zero chemical potential as well as in the symmetric case of Δ = t, and by using density-matrix-renormalization-group method for interacting system with nearest neighbor repulsion interaction. We suggest and examine an edge correlation function of Majorana fermions to characterize the long range order in the topological superconducting states and study the phase diagram of the interating Kitaev chain.
Majorana modes can arise as zero energy bound states in a variety of solid state systems. A lattice of these quasiparticles, for instance, emerges on the surface of a topological superconductor with the zero modes localized at the cores of vortices. This thesis is devoted to the study of interactions between the Majorana zero modes when such a lattice has a triangular geometry. We map the phase diagram of this model using a combination of mean field theory and numerical simulation of ladders through the density-matrix renormalization-group technique. Our analysis suggests that interactions drive a topologically non-trivial gapped phase into a gapless Luttinger liquid.
Majorana modes can arise as zero energy bound states in a variety of solid state systems. A lattice of these quasiparticles, for instance, emerges on the surface of a topological superconductor with the zero modes localized at the cores of vortices. This thesis is devoted to the study of interactions between the Majorana zero modes when such a lattice has a triangular geometry. We map the phase diagram of this model using a combination of mean field theory and numerical simulation of ladders through the density-matrix renormalization-group technique. Our analysis suggests that interactions drive a topologically non-trivial gapped phase into a gapless Luttinger liquid.
We revisit the phase diagram of the dimerized XXZ spin-
21 chain with nearest-neighbor couplings which was studied numerically by Mondal et al. [Phys. Rev. B 106, L201106 (2022)]. The model has isotropic
XY couplings which have a uniform value and
ZZ couplings which have a dimerized form, with strengths
Ja and
Jb on alternate bonds. We find a rich phase diagram in the region of positive
Ja,Jb. We provide a detailed understanding of the different phases and associated quantum phase transitions using a combination of mean-field theory, low-energy effective Hamiltonians, renormalization group calculations employing the technique of bosonization, and numerical calculations using the density-matrix renormalization group (DMRG) method. The phase diagram consists of two Ising paramagnetic phases called
IPM0 and ${\mathrm{IPM}}_{\ensuremath{\pi}}$, and a phase with Ising Neel (IN) order; all these phases are gapped. The phases
IPM0 and ${\mathrm{IPM}}_{\ensuremath{\pi}}$ are separated by a gapless phase transition line $0\ensuremath{\le}{J}_{a}={J}_{b}\ensuremath{\le}1$---the well-known Tomonaga Luttinger liquid captured by a conformal field theory (CFT) with central charge
c=1. In contrast, there are two phase transition lines separating
IPM0 or the ${\mathrm{IPM}}_{\ensuremath{\pi}}$ from IN that are described by
c=21 CFTs and correspond to quantum Ising transitions. In fact, the
c=1 line bifurcates into the two
c=21 lines at the point
Ja=Jb=1; the shape of the bifurcation is found analytically using RG calculations. A symmetry analysis shows that
IPM0 is a topologically trivial phase while ${\mathrm{IPM}}_{\ensuremath{\pi}}$ is a time-reversal symmetry-protected topological phase (SPT) with spin-
21 states at the two ends of an open system. A gap opens in the low-energy spectrum whenever one moves away from one of the phase transition lines; the scaling of the gap with the distance from a transition line is found analytically using the RG method. The numerical results obtained by the DMRG method are in good agreement with the analytical results. Further, we show that useful insights into the nature of the phase diagram of the above Hamiltonian, particularly the SPT as well as the phase transitions, is obtained via Kramers-Wannier duality to a deformed quantum Ashkin-Teller model. Finally, we propose experimental platforms for testing our results.