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arXiv:2308.01494 · Physical review. B./Physical review. B

Polarity of the fermionic condensation in the p -wave Kitaev model on a square lattice

E. S. Ma, Z. Song

Published Nov 28, 2023• 2 citations
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Abstract

In a $p$-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 $p$-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.

DOI
10.1103/physrevb.108.195150

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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.
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Motivated by the work of Kitaev, we construct an exactly soluble spin-½ model on a honeycomb lattice whose ground states are identical to Δ 1 x p x +Δ 1 y p y + i (Δ 2 x p x +Δ 2 y p y )-wave paired fermions on a square lattice, with tunable paring order parameters. We derive a universal phase diagram for this general p -wave theory which contains a gapped A-phase and a topologically non-trivial B-phase. We show that the gapless condition in the B-phase is governed by a generalized inversion (G-inversion) symmetry under p x ↔ (Δ 1 y / Δ 1 x ) p y . The G-inversion symmetric gapless B-phase near the phase boundaries is described by (1+1)-dimensional gapless Majorana fermions in the asymptotic long-wavelength limit, i.e. the c =1/2 conformal field theory. The gapped B-phase has G-inversion symmetry breaking and is the weak pairing phase described by the Moore-Read Pfaffian. We show that in the gapped B-phase, vortex pair excitations are separated from the ground state by a finite energy gap.
We study a one-dimensional Kitaev model with uniform phase gradient pairing term. We show that the gradient constant significantly affects the phase diagram, which consists of topologically trivial and nontrivial phases, associated with Majorana edge modes. Based on the exact solution, a Bardeen-Cooper-Schrieffer-pair order parameter is introduced to characterize the phase diagram by its value and nonanalytic behavior at phase boundaries. We find that this order parameter obtains its maxima at the triple critical points, at which the pairing phase gradient suppresses the single-particle scattering process due to the coherent destructive interference. In particular, we show that the ground state at such a point possesses exact off-diagonal long-range order (ODLRO), in the thermodynamic limit. Our result provides an example of a gapless ppp-wave superconducting ground state possessing ODLRO.
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.
We determine the relative stability of different ground-state phases of spin-imbalanced populations of attractive fermions in square lattices. The ground state is determined within Hartree-Fock-Bogoliubov theory, with care taken to remove finite-size effects. The phases are systematically characterized by the symmetry of the order parameter and their real- and momentum-space structures. For quarter- to half-filled lattices, where the Fermi surfaces are most distorted from their spherical counterpart in the continuum, we predominantly find unidirectional Larkin-Ovchinikov-type phases. We discuss the effect of commensuration between the ordering wave vector and the density imbalance and describe the mechanism of Fermi surface reconstruction and pairing for various orders. A robust supersolid phase exists when the ordering wave vector is diagonally directed. Charge and pairing order coexist, rather than compete, and are responsible for the opening of the gap on different portions of the Fermi surface. A variational determination of the correct pair momentum of the Larkin-Ovchinikov phases shows that phase separation does not occur in the considered regime of density and magnetization.