Within the framework of a one-dimensional model of interacting electrons, the ground state of an electron liquid is studied. Using the exact solution of the model, the ground state phase diagram and zero-energy Majorana edge functions in a finite chain are calculated. The winding number invariant reflects the topological nature of the electron liquid. The phase diagram includes two topological phases with different winding number invariants, the topologically trivial Mott insulator phase, and three critical phase transition points. Numerical calculations confirm and illustrate the analytical results.
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.
We investigate properties of a topological Mott insulator in one dimension by examining the bulk topological invariant and the entanglement spectrum of a correlated electron model. We clarify how gapless edge states in a noninteracting topological band insulator evolve into spinon edge states in a topological Mott insulator. Furthermore, we propose a topological Mott transition, which is a new type of topological phase transition which has never been observed in free fermion systems. This unconventional transition occurs in spin liquid phases in the Mott insulator and is accompanied by zeros of the single-electron Green's function and a gap closing in the spin excitation spectrum.
We study the ground-state phase diagram and dynamics of the one-dimensional cluster model with several competing interactions. Paying particular attention to the relation between the entanglement spectrum (ES) and the bulk topological (winding) number, we first map out the ground-state phases of the model and determine the universality classes of the transitions from the exact solution. We then investigate the dynamical properties during interaction sweeps through the critical points of topological phase transitions. When the sweep speed is slow, the correlation functions and the entanglement entropy exhibit spatially periodic structures. On top of this, the levels in the ES oscillate temporally during the dynamics. By explicitly calculating the above quantities for excited states, we attribute these behaviors to the Bogoliubov quasiparticles generated near the critical points. We also show that the ES reflects the strength of the Majorana correlation even for the excited states.
We study the phases of an exactly solvable one dimensional model with
4−dimensional
Γ−matrix degrees of freedom on each site. The
Γ−matrix model has a large set of competing interactions and displays a rich phase diagram with critical lines and multi-critical points. We work with the model with certain
Z2 symmetries and identify the allowed symmetry protected topological phases using the winding number as the topological invariant. The model belongs to the CII-class of the
10−fold classification and allows for integer values of the winding number. We confirm that the system also hosts localized zero energy Majorana edge modes, consistent with the integer value of the winding number of the corresponding phase. We further study scaling and universality behaviour of the various topological phase transitions.
We investigate the spectral and topological properties of a zigzag Kitaev chain constructed from two diagonally coupled one dimensional Kitaev chains with a zero and finite superconducting pairing phase difference. Using a Bogoliubov-de Gennes formulation, we analyze the energy spectrum, distribution of Majorana zero modes (MZMs), the quasi-particle dispersion, and the winding number, respectively. For a zero phase difference, the resulting energy spectrum shows topological phases with two, four MZMs, and trivial regions. The phases of gap closure determine the topological phase boundaries. In particular, for the phase difference between
φ=π, the degeneracy of MZMs is partially lifted, leading to modified topological phases compared to the case
φ=0. The topological and trivial phase boundaries are further confirmed by evaluating the quasi-particle dispersion and the topological invariant, namely the winding number. We show that the zigzag Kitaev chain contributes independently to the total winding number
ν=1 and
2, giving rise to distinct topological phases that support two and four MZMs. The
ν=0 gives rise to a trivial region. The energy spectrum of systems corroborates the analytical phase boundaries and reveals characteristics associated with hybridization, enabling us to obtain the complete phase diagram of the zigzag model. Our results establish the zigzag Kitaev chain as a minimal platform for engineering MZM quantum computations, with potential applications in the study of topological phases and Majorana based qubit physics.