Topological and quantum critical properties of the interacting Majorana chain model

Abstract

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}_2ℤ2 with broken parity symmetry and the period-2-\mathbb{Z}_2ℤ2 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}_2ℤ2 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.

Ground-state phase diagram and universality of sequential topological valence-bond-solid quantum transitions in a mixed tetramer chain

“…In the Luttinger spin-liquid phase, the transverse correlation function should decay according to a power law G(r) ∝ 1/r 1/2K due to its critical nature [1,47]. The Luttinger parameter K determines the existence of Tomonaga-Luttinger spin liquid regime for specific values at the interval 1 < K < 2 [48,49]. The transverse spin-spin correlation within the DMRG method implementing MPS is straightforwardly calculated as a mean value of the product of the relevant raising and lowering spin operators.…”

“…implemented to identify and characterize the Luttinger regime in the hard-core Bose model with three-body interactions [48] and interacting Majorana chains [49].…”

Critical properties of the Majorana chain with competing interactions

“…At the same time, and quite logically since fermions experience a repulsion, theoretical studies of Majorana chains are centered on extended models incorporating interactions between Majorana fermions in various forms [17][18][19][20][21][22][23][24][25][26]. In this paper we study the combined effect of the two interaction terms of the shortest possible range.…”

“…The Majorana chain with h = t = 1 and g interaction only has been intensely studied in recent years [18,19,26,33]. It was shown that for small coupling g the system remains in the critical Ising phase, however, at g ≈ 0.29 it undergoes a Lifshitz transition into a critical phase with the central charge c = 3 2 [19,26]. This phase is the Ising critical phase (c = 1 2 ) superposed with the floating phase, Luttinger liquid phase (c = 1), with incommensurate correlations.…”

“…Beyond this terminal point the system was expected to be gapped with fourfold degenerate ground state [19,33]. However, in the recent study of an extended phase diagram with J = h [26] it was shown that the floating phase terminates much earlier, at g ≈ 1.3, the Ising criticality terminates at g ≈ 3 with the tricritical Ising point, beyond which the system is indeed gapped, but the ground state is sixfold degenerate. It was also shown that incommensurability persists beyond g = 3.…”