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arXiv:2310.05006 · New Journal of Physics

Real-space decomposition of p-wave Kitaev chain

D. K. He, E. S. Ma, Z. Song

Published Jan 30, 2024• 0 citations
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Abstract

Abstract We propose an extended Bogoliubov transformation in real space for spinless fermions, based on which a class of Kitaev chains of length 2N with zero chemical potential can be mapped to two independent Kitaev chains of length N. It provides an alternative way to investigate a complicated system from the result of relatively simple systems. We demonstrate the implications of this decomposition by a Su–Schrieffer–Heeger Kitaev model, which supports rich quantum phases. The features of the system, including the groundstate topology and nonequilibrium dynamics, can be revealed directly from that of sub-Kitaev chains. Based on this connection, two types of Bardeen–Cooper–Schrieffer (BCS)-pair order parameters are introduced to characterize the phase diagram, showing the ingredient of two different BCS pairing modes. Analytical analysis and numerical simulations show that the real-space decomposition for the ground state still holds true approximately in presence of finite chemical potential in the gapful regions.

DOI
10.1088/1367-2630/ad2412

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Abstract We propose an extended Bogoliubov transformation in real space for spinless fermions, based on which a class of Kitaev chains of length 2N with zero chemical potential can be mapped to two independent Kitaev chains of length N. It provides an alternative way to investigate a complicated system from the result of relatively simple systems. We demonstrate the implications of this decomposition by a Su–Schrieffer–Heeger Kitaev model, which supports rich quantum phases. The features of the system, including the groundstate topology and nonequilibrium dynamics, can be revealed directly from that of sub-Kitaev chains. Based on this connection, two types of Bardeen–Cooper–Schrieffer (BCS)-pair order parameters are introduced to characterize the phase diagram, showing the ingredient of two different BCS pairing modes. Analytical analysis and numerical simulations show that the real-space decomposition for the ground state still holds true approximately in presence of finite chemical potential in the gapful regions.
We introduce a frustration-free, one-dimensional model of spinless fermions with hopping, ppp-wave superconducting pairing and alternating chemical potentials. The model possesses two exactly degenerate ground states even for finite system sizes. We present analytical results for the strong Majorana zero modes, the phase diagram, and the topological order. Furthermore, we generalize our results to include interactions.
In a ppp-wave Kitaev model, the nearest-neighbor pairing term results in the formation of the BCS pair in the ground state. In this work, we study the fermionic condensation of real-space pairs in a ppp-wave Kitaev model on a square lattice with a uniform phase gradient pairing term along both directions. The exact solution shows that the ground state can be expressed in a coherent-state-like form, indicating the condensation of a collective pairing mode, which is the superposition of different configurations of pairs in real space. The amplitudes of each configuration depend not only on the size but also on the orientation of the pair. We employ three quantities to characterize the ground state in the thermodynamic limit. (i) A BCS-pair order parameter is introduced to characterize the phase diagram, consisting of gapful and topological gapless phases. (ii) The particle-particle correlation length is obtained to reveal the polarity of the pair condensation. (iii) In addition, a pair-pair correlator is analytically derived to indicate the presence of off-diagonal long-range order. Our work proposes an alternative method for understanding fermionic condensation.
A nonequilibrium state shares macroscopic properties, such as conductivity and superconductivity, with a static state when the two states exhibit an identical average value of an observable over a period. We studied the quench dynamics of a Kitaev chain on the basis of two types of order parameters associated with two channels of pairing: local pairing in real space and Bardeen-Cooper-Schrieffer (BCS)-like pairing in momentum space. Based on the exact solution, we found that the two order parameters were identical for the ground state, which indicated a balance between the two pairing channels and would help determine the quantum phase diagram. However, for a nonequilibrium state obtained through time evolution from an initially prepared vacuum state, the two parameters varied, but both parameters could still help determine the phase diagram. In the region of nontrivial topological phases, nonequilibrium states favored BCS-like pairing. In this paper, we present an alternative approach to dynamically generate a superconducting state from a trivial empty state and illuminate the pairing mechanism.
Abstract We study the topological bulk and edge correlations of condensed ground state in a p-wave Kitaev model on a square lattice. It is shown that the ground state of the system has the form of the condensate of the Bardeen-Cooper-Schrieffer pair, and the topological transition is associated with the nonanalytic behavior of the pairing order parameters. A real space correlation function is proposed to characterizing the topological phase of the many-body ground state. Numerical results demonstrate that this method works well in the presence of disordered perturbation, lattice defects, or irregular boundary conditions. In addition, the real space correlation function between two edges of the system is investigated, which directly reflects the existence of topological edge modes in the many-body ground state.
We study a system of interacting spinless fermions in one dimension that, in the absence of interactions, reduces to the Kitaev chain [Kitaev, Phys. Usp. 44, 131 (2001)]. In the noninteracting case, a signal of topological order appears as zero-energy modes localized near the edges. We show that the exact ground states can be obtained analytically even in the presence of nearest-neighbor repulsive interactions when the on-site (chemical) potential is tuned to a particular function of the other parameters. As with the noninteracting case, the obtained ground states are twofold degenerate and differ in fermionic parity. We prove the uniqueness of the obtained ground states and show that they can be continuously deformed to the ground states of the noninteracting Kitaev chain without gap closing. We also demonstrate explicitly that there exists a set of operators each of which maps one of the ground states to the other with opposite fermionic parity. These operators can be thought of as an interacting generalization of Majorana edge zero modes.
The fermionic and bosonic zero modes of the one-dimensional (1D) interacting Kitaev chain at the symmetric point are unveiled. The many-body structures of the Majorana zero modes in the topological region are given explicitly by carrying out a perturbation expansion up to infinite order. We also give the analytic expressions of the bosonic zero modes in the topologically trivial phase. Our results are generalized to the hybrid fermion system comprised of the interacting Kitaev model and the Su–Schrieffer–Heeger (SSH) model, in which we show that these two types of zero modes can coexist in a certain region of its phase diagram.