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arXiv:2204.06306 · Physical review. B./Physical review. B

First-order topological phase transitions and disorder-induced Majorana modes in interacting fermion chains

Shruti Agarwal, Shreekant Gawande, Satoshi Nishimoto, Jeroen van den Brink, Sanjeev Kumar

Published Mar 20, 2023• 1 citation
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Abstract

Using a combination of the mean-field Bogoliubov--de Gennes approach and the density matrix renormalization group method, we discover first-order topological transitions between topological superconducting and trivial insulating phases in a sawtooth lattice of intersite attractive fermions. The topological characterization of different phases is achieved in terms of winding numbers, Majorana edge modes, and entanglement spectra. By studying the effect of disorder on first-order topological phase transitions, we establish disorder-induced topological phase coexistence as a mechanism for generating a finite density of Majorana particles.

DOI
10.1103/physrevb.107.l121106

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Using a combination of the mean-field Bogoliubov--de Gennes approach and the density matrix renormalization group method, we discover first-order topological transitions between topological superconducting and trivial insulating phases in a sawtooth lattice of intersite attractive fermions. The topological characterization of different phases is achieved in terms of winding numbers, Majorana edge modes, and entanglement spectra. By studying the effect of disorder on first-order topological phase transitions, we establish disorder-induced topological phase coexistence as a mechanism for generating a finite density of Majorana particles.
We study one-component fermions in chain lattices with proximity-induced superconducting gap and interparticle short-range interaction, capable of hosting Majorana fermions. By systematically tracking various physical quantities, we show that topological states and topological phase transitions in the system can be identified by multiple signatures in thermodynamic quantities and pair-condensate properties, in good agreement with the known signatures in the ground-state energy and entanglement spectrum. We find the disappearance of the topological phase in a largely attractive regime, in which the system undergoes a first-order transition between two topologically trivial states. In addition, the stability of the signatures against finite size, disorder, and inhomogeneity is analyzed. Our results provide additional degrees of freedom for the characterization of topological states with interaction and for the experimental detection of emergent Majorana fermions.
We study a periodically driven one-dimensional Kitaev model in the presence of disorder. In the clean limit our model exhibits four topological phases corresponding to the existence or nonexistence of edge modes at zero and $\ensuremath{\pi}$ quasienergy. When potential disorder is added, the system parameters get renormalized and the system may exhibit a topological phase transition. When starting from the Majorana $\ensuremath{\pi}$ mode (MPM) phase, which hosts only edge Majoranas with quasienergy $\ensuremath{\pi}$, disorder induces a transition into a neighboring phase with both $\ensuremath{\pi}$ and zero modes on the edges. We characterize the disordered system using (i) exact diagonalization, (ii) Arnoldi mapping onto an effective tight-binding chain, and (iii) topological entanglement entropy.
The Hubbard and spinless fermion chains are paradigms of strongly correlated systems, very well understood using the 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, an 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 four 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 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 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.
Among the broad spectrum of systems predicted to exhibit topological superconductivity and Majorana fermions, one-dimensional wires with strong spin-orbit coupling provide one of the most promising experimental candidates. Here we investigate the fate of the topological superconducting phase in such wires when repulsive interactions are present. Using a combination of density matrix renormalization group, bosonization, and Hartree--Fock techniques, we demonstrate that while interactions degrade the bulk gap---consistent with recent results of Gangadharaiah et al.---they also greatly expand the parameter range over which the topological phase arises. In particular, we show that with interactions this phase can be accessed over a broader chemical potential window, thereby leading to greater immunity against disorder-induced chemical potential fluctuations in the wire. We also suggest that in certain wires strong interactions may allow Majorana fermions to be generated without requiring a magnetic field.
We study the combined effect of interactions and disorder on topological order in one dimension. Toward that end, we consider a generalized Kitaev chain including fermion-fermion interactions and disorder in the chemical potential. We determine the phase diagram by performing density-matrix renormalization-group calculations on the corresponding spin-1/2 chain. We find that moderate disorder or repulsive interactions individually stabilize the topological order, which remains valid for their combined effect. However, both repulsive and attractive interactions lead to a suppression of the topological phase at strong disorder.
We elucidate the reduction of the winding number caused by the on-site disorder in a next-nearest-neighbor XY model. When disorder becomes strong enough, Majorana edge modes become critically extended, beyond which they collapse into Anderson localized (AL) states in the bulk, resulting in a topological Anderson insulating state. We identify a resilience threshold Wt{W}_{t}Wt​ for every pair of Majorana fermions (MFs). In response to increasing disorder, every pair of MFs collapse into AL states in the bulk beyond their resilience threshold. For very strong disorder, all Majorana fermions collapse, and a topologically trivial state is obtained. We show that the threshold values are related to the localization length of Majorana fermions, which can be efficiently calculated by an appropriate modification of the transfer-matrix method. At the topological transition point, the localization length of the zero modes diverges, and the system becomes scale invariant. The number of peaks in the localization length as a function of disorder strength determines the number of zero modes in the clean state before disorder is introduced. This finding elevates the transfer-matrix method to the level of a tool for the determination of the topological index of both clean and disordered systems.