Phase Diagram of the Interacting Majorana Chain Model

TL;DR

This work analyzes a minimal one-dimensional model of interacting Majorana zero modes with four-site interactions, revealing a rich phase diagram that includes a gapless Ising phase, a gapless Ising+LL phase with emergent U(1) charge, a Lifshitz transition with dynamical exponent z=3, and a large-|g| gapped phase with 4-fold degeneracy. It uncovers a tricritical Ising CFT at g>0 signaling emergent supersymmetry, and introduces a generalized commensurate-incommensurate transition that couples Ising and Luttinger-liquid sectors. A self-consistent mean-field treatment of dimerized versions maps out topological gapped phases labeled by edge Majorana modes, while DMRG confirms key features and provides detailed finite-size scalings. The results offer a comprehensive theoretical framework for realizing and probing interacting Majorana chains in 1D experimental platforms, with distinctive tunneling signatures across the various phases.

Abstract

The Hubbard chain and spinless fermion chain are paradigms of strongly correlated systems, very well understood using Bethe ansatz, Density Matrix Renormalization Group (DMRG) and field theory/renormalization group (RG) methods. They have been applied to one-dimensional materials and have provided important insights for understanding higher dimensional cases. Recently, a new interacting fermion model has been introduced, with possible applications to topological materials. It has a single Majorana fermion operator on each lattice site and interactions with the shortest possible range that involve 4 sites. We present a thorough analysis of the phase diagram of this model in one dimension using field theory/RG and DMRG methods. It includes a gapped supersymmetric region and a novel gapless phase with coexisting Luttinger liquid and Ising degrees of freedom. In addition to a first order transition, three critical points occur: tricritical Ising, Lifshitz and a novel generalization of the commensurate-incommensurate transition. We also survey various gapped phases of the system that arise when the translation symmetry is broken by dimerization and find both trivial and topological phases with 0, 1 and 2 Majorana zero modes bound to the edges of the chain with open boundary conditions.

Figures (13)

• Figure 1: (a) The Hubbard chain with on-site interactions. (b) The spinless Dirac chain with nearest-neighbor interactions. (c) The most local Majorana chain with 4-site interactions. (d) The phase diagram of the Majorana chain with the Hamiltonian \ref{['eq:hamil']} as a function of $g/t$. Setting $t=1$, it consists of four phases: a 4-fold degenerate gapped phase separated by a generalized commensurate-incommensurate (C-IC) transition at $g\approx -2.86$ from a critical phase with central charge $c=3/2$ comprised of a critical Ising and a decoupled Luttinger liquid (LL). The Ising+LL phase is, in turn, separated from a critical Ising phase by a Lifshitz critical point with dynamical exponent $z=3$ at $g\approx -0.28$. For positive $g$, we have another transition from the Ising phase to a doubly degenerate supersymmetric gapped phase at $g\approx 250$ with the phase transition described by the tricritical Ising (TCI) conformal field theory (CFT) with central charge $c=7/10$.
• Figure 2: Mean-field ground states for $g>0$, corresponding to ferromagnetic states in the spin representation. The small blue and green circles represent the sites of the Majorana chain. Bold links with large circles on them represent Dirac fermions formed by combining two Majoranas (two types of combinations are considered: blue-green and green-blue). A filled circle corresponds to the Dirac level being filled and an empty circle to it being empty.
• Figure 3: Mean-field ground states for $g<0$, corresponding to the antiferromagnetic states in the spin representation
• Figure 4: Velocity in the Ising phase near the Lifshitz transition. As the transition is driven by a renormalization of the dispersion relation, mean-field calculations are in approximate agreement with DMRG.
• Figure 5: The behavior of the central charge as a function of $t$ for $g=-1$.