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arXiv:2202.11450 · Condensed Matter

Topological Phases of an Interacting Majorana Benalcazar–Bernevig–Hughes Model

Alfonso Maiellaro, Fabrizio Illuminati, R. Citro

Published Mar 4, 2022• 10 citations
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Abstract

We study the effects of Coulomb repulsive interactions on a Majorana Benalcazar–Bernevig–Huges (MBBH) model. The MBBH model belongs to the class of second-order topological superconductors (HOTSC2), featuring robust Majorana corner modes. We consider an interacting strip of four chains of length L and perform a density matrix renormalization group (DMRG) numerical simulation based on a tensor-network approach. Study of the non-local fermionic correlations and the degenerate entanglement spectrum indicates that the topological phases are robust in the presence of interactions, even in the strongly interacting regime.

DOI
10.3390/condmat7010026

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We study the effects of Coulomb repulsive interactions on a Majorana Benalcazar–Bernevig–Huges (MBBH) model. The MBBH model belongs to the class of second-order topological superconductors (HOTSC2), featuring robust Majorana corner modes. We consider an interacting strip of four chains of length L and perform a density matrix renormalization group (DMRG) numerical simulation based on a tensor-network approach. Study of the non-local fermionic correlations and the degenerate entanglement spectrum indicates that the topological phases are robust in the presence of interactions, even in the strongly interacting regime.
In this work, we study the topological phase transitions of a Kitaev chain generalized by the addition of nearest-neighbor Coulomb interaction. We show the presence of a robust topological phase as a function of the interaction strength and of the on-site energy with associated non-zero energy Majorana states localized at the chain edges. We provide an effective mean-field model that allows for the self-consistent computation of the mean value of the local particle number operator, and we also perform Density Matrix Renormalization Group numerical simulations based on a tensor network approach. We find that the two methods show a good agreement in reporting the phase transition between trivial and topological superconductivity. Temperature robustness within a physically relevant threshold has also been demonstrated. These findings shed light on an entire class of topological interacting one-dimensional systems in which the effects of residual Coulomb interactions play a relevant role.
In this study, the problem of strong Coulomb interactions in topological superconducting wire is analyzed by means of the density-matrix-renormalization-group (DMRG) approach. To analyze properties of edge states in the BDI-class structure, a quantity called Majorana polarization is used. From its dependence on wire length and an entanglement-spectrum degeneracy, topological phase diagrams are obtained. The DMRG calculations for the Shubin-Vonsovsky--type model of the wire show the transformation of phases with Majorana single and double modes (MSMs and MDMs, respectively) under the increase of onsite and intersite correlations. In particular, we demonstrate different scenarios including the possibilities of both induction and suppression of the MSMs and MDMs. It is shown that in the strongly correlated regime, the contributions of single-particle excitations to the Majorana-type states significantly decrease at low magnetic fxields. Moreover, the $t\ensuremath{-}{J}^{*}\ensuremath{-}V$ model is derived allowing to study the effective interactions and improve the DMRG numerics. It is found out that in the limiting case of the effective Hamiltonian with infinitely strong onsite repulsion, ttt model, the topological phases are destroyed. Finally, the ways to probe the MSMs and MDMs via the features of caloric functions are discussed.
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.
A superconducting wire described by a ppp-wave pairing and a Kitaev Hamiltonian exhibits Majorana fermions at its edges and is topologically protected by symmetry. We consider two Kitaev wires (chains) coupled by a Coulomb-type interaction and study the complete phase diagram using analytical and numerical techniques. A topological superconducting phase with four Majorana fermions occurs until moderate interactions between chains. For large interactions, both repulsive and attractive, by analogy with the Hubbard model, we identify Mott phases with Ising-type magnetic order. For repulsive interactions, the Ising antiferromagnetic order favors the occurrence of orbital currents spontaneously breaking time-reversal symmetry. By strongly varying the chemical potentials of the two chains, quantum phase transitions towards fully polarized (empty or full) fermionic chains occur. In the Kitaev model, the quantum critical point separating the topological superconducting phase and the polarized phase belongs to the universality class of the critical Ising model in two dimensions. When increasing the Coulomb interaction between chains, then we identify an additional phase corresponding to two critical Ising theories (or two chains of Majorana fermions). We confirm the existence of such a phase from exact mappings and from the concept of bipartite fluctuations. We show the existence of negative logarithmic corrections in the bipartite fluctuations, as a reminiscence of the quantum critical point in the Kitaev model. Other entanglement probes such as bipartite entropy and entanglement spectrum are also used to characterize the phase diagram. The limit of large interactions can be reached in an equivalent setup of ultracold atoms and Josephson junctions.
We investigate the effects of electronic correlations on the Bernevig-Hughes-Zhang model using the real-space density matrix renormalization group (DMRG) algorithm. We introduce a method to probe topological phase transitions in systems with strong correlations using DMRG, substantiated by an unsupervised machine learning methodology that analyzes the orbital structure of the real-space edges. Including the full multi-orbital Hubbard interaction term, we construct a phase diagram as a function of a gap parameter (mmm) and the Hubbard interaction strength (UUU) via exact DMRG simulations on $N\ifmmode\times\else\texttimes\fi{}4$ cylinders. Our analysis confirms that the topological phase persists in the presence of interactions, consistent with previous studies, but it also reveals an intriguing phase transition from a paramagnetic to a stripey antiferromagnetic topological insulator. The combination of the magnetic structure factor, strength of magnetic moments, and the orbitally resolved density, provides real-space information on both topology and magnetism in a strongly correlated system.
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.