Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physica Scripta

Topological bulk and edge correlations in a Kitaev model on a square lattice

E. S. Ma, K. L. Zhang, Z. Song

Published Feb 7, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/adb3d6

Related papers

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

E. S. Ma, Z. Song

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.

arXiv:2308.01494•Physical review. B./Physical review. B•Nov 28, 2023•2 citations

Emergence of new topological gapless phases in the modified square-lattice Kitaev model

Jihyeon Park, Gun Sang Jeon

We investigate emergent topological gapless phases in the square-lattice Kitaev model with additional hopping terms. In the presence of nearest-neighbor hopping only, the model is known to exhibit gapless phases with two topological gapless points. When the strength of the newly added next-nearest-neighbor hopping is smaller than a certain value, qualitatively the same phase diagram persists. We find that further increase of the extra hopping results in a new topological phase with four gapless points. We construct a phase diagram to clarify the regions of emergent topological gapless phases as well as topologically trivial ones in the space of the chemical potential and the next-nearest-neighbor hopping strength. We examine the evolution of the gapless phases in the energy dispersions of the bulk as the chemical potential varies. The topological properties of the gapless phases are characterized by the winding numbers of the present gapless points. We also consider the ribbon geometry to examine the corresponding topological edge states. It is revealed that Majorana-fermion edge modes exist as flat bands in topological gapless phases. We also perform the analytical calculation as to Majorana-fermion zero-energy modes and discuss its implications on the numerical results.

arXiv:2311.11684•Nov 20, 2023•0 citations

Off-diagonal long-range order in the ground state of the Kitaev chain

E. S. Ma, Z. Song

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 $p$-wave superconducting ground state possessing ODLRO.

arXiv:2212.02694•Physical review. B./Physical review. B•May 10, 2023•9 citations

Singularities in ground-state fidelity and quantum phase transitions for the Kitaev model

Jian-Hui Zhao, Huan-Qiang Zhou

The ground-state fidelity per lattice site is shown to be able to detect quantum phase transitions for the Kitaev model on the honeycomb lattice, a prototypical example of quantum lattice systems with topological order. It is found that, in the thermodynamic limit, the ground-state fidelity per lattice site is nonanalytic at the phase boundaries; the second-order derivative of its logarithmic function with respect to a control parameter describing the interaction between neighboring spins is divergent. A finite-size scaling analysis is performed, which allows us to extract the correlation length critical exponent from the scaling behaviors of a part of the ground-state fidelity per lattice site.

arXiv:0803.0814•Physical Review B•Jul 1, 2009•58 citations

Topological edge states in the spin 1 bilinear–biquadratic model

Peng Li, Su-Peng Kou

The spin 1 bilinear-biquadratic model H = ∑(ij)[cosφS(i)·S(j) + sinφ(S(i)·S(j))(2)] on a square lattice in the region 0 < φ < π/4 is studied in a fermion representation with a p-wave pairing Bardeen-Cooper-Schrieffer type of mean-field theory. Our results show there may exist a non-trivial gapped spin liquid with time-reversal symmetry spontaneously breaking. This exotic state manifests its topological nature by forming chiral states at the edges. To show this more clearly, we set up and solved a ribbon system. We got a gapless dispersion representing the edge modes beneath the bulk modes. The edge modes with nonzero longitudinal momentum (k(x) ≠ 0) convect in opposite directions at the two edges, which leads to a twofold degeneracy, while the modes with zero longitudinal momentum (k(x) = 0) turn out to be Majorana fermion states. The edge spin correlation functions are found to decay following a power law with increasing distance. We also calculated the contribution of the edge modes to the specific heat and obtained a linear law at low temperatures.

arXiv:1211.0892•Journal of Physics Condensed Matter•Oct 10, 2012•5 citations

Quantum phase transitions in the Kitaev model on decorated lattices

Igor N. Karnaukhov

We have proposed an exactly solvable model defined on 2D decorated lattices of two types. The ground-state phase diagram of the system includes different topological phases with gapless chiral edge states. We show that two types of chiral spin liquid with gapless edge modes are realized on lattices with different symmetry. The phase transition between the topological phase with chiral gapped (Chern number zero) and the topological phase with chiral gapless edge modes (Chern number ) occurs in the model on the square (symmetric) decorated lattice. On the rectangular (asymmetric) decorated lattice the topological phase is defined by a chiral gapless (gapped) edge mode in the x ( y ) direction and a chiral gapped (gapless) edge mode in another y ( x ) direction. We show that a Kitaev model on a decorated asymmetric square lattice exhibits the quantum phase transition between topological phases with equal Chern numbers.

Europhysics Letters (EPL)•Jun 1, 2013•3 citations
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.
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.
We investigate emergent topological gapless phases in the square-lattice Kitaev model with additional hopping terms. In the presence of nearest-neighbor hopping only, the model is known to exhibit gapless phases with two topological gapless points. When the strength of the newly added next-nearest-neighbor hopping is smaller than a certain value, qualitatively the same phase diagram persists. We find that further increase of the extra hopping results in a new topological phase with four gapless points. We construct a phase diagram to clarify the regions of emergent topological gapless phases as well as topologically trivial ones in the space of the chemical potential and the next-nearest-neighbor hopping strength. We examine the evolution of the gapless phases in the energy dispersions of the bulk as the chemical potential varies. The topological properties of the gapless phases are characterized by the winding numbers of the present gapless points. We also consider the ribbon geometry to examine the corresponding topological edge states. It is revealed that Majorana-fermion edge modes exist as flat bands in topological gapless phases. We also perform the analytical calculation as to Majorana-fermion zero-energy modes and discuss its implications on the numerical results.
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.
The ground-state fidelity per lattice site is shown to be able to detect quantum phase transitions for the Kitaev model on the honeycomb lattice, a prototypical example of quantum lattice systems with topological order. It is found that, in the thermodynamic limit, the ground-state fidelity per lattice site is nonanalytic at the phase boundaries; the second-order derivative of its logarithmic function with respect to a control parameter describing the interaction between neighboring spins is divergent. A finite-size scaling analysis is performed, which allows us to extract the correlation length critical exponent from the scaling behaviors of a part of the ground-state fidelity per lattice site.
The spin 1 bilinear-biquadratic model H = ∑(ij)[cosφS(i)·S(j) + sinφ(S(i)·S(j))(2)] on a square lattice in the region 0 < φ < π/4 is studied in a fermion representation with a p-wave pairing Bardeen-Cooper-Schrieffer type of mean-field theory. Our results show there may exist a non-trivial gapped spin liquid with time-reversal symmetry spontaneously breaking. This exotic state manifests its topological nature by forming chiral states at the edges. To show this more clearly, we set up and solved a ribbon system. We got a gapless dispersion representing the edge modes beneath the bulk modes. The edge modes with nonzero longitudinal momentum (k(x) ≠ 0) convect in opposite directions at the two edges, which leads to a twofold degeneracy, while the modes with zero longitudinal momentum (k(x) = 0) turn out to be Majorana fermion states. The edge spin correlation functions are found to decay following a power law with increasing distance. We also calculated the contribution of the edge modes to the specific heat and obtained a linear law at low temperatures.
We have proposed an exactly solvable model defined on 2D decorated lattices of two types. The ground-state phase diagram of the system includes different topological phases with gapless chiral edge states. We show that two types of chiral spin liquid with gapless edge modes are realized on lattices with different symmetry. The phase transition between the topological phase with chiral gapped (Chern number zero) and the topological phase with chiral gapless edge modes (Chern number ) occurs in the model on the square (symmetric) decorated lattice. On the rectangular (asymmetric) decorated lattice the topological phase is defined by a chiral gapless (gapped) edge mode in the x ( y ) direction and a chiral gapped (gapless) edge mode in another y ( x ) direction. We show that a Kitaev model on a decorated asymmetric square lattice exhibits the quantum phase transition between topological phases with equal Chern numbers.