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

arXiv:1608.08382 · Scientific Reports

Majorana zero modes and long range edge correlation in interacting Kitaev chains: analytic solutions and density-matrix-renormalization-group study

Jian-Jian Miao, Hui-Ke Jin, Fu‐Chun Zhang, Yi Zhou

Published Jan 5, 2018• 25 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1038/s41598-017-17699-y

Related papers

Majorana zero modes, spontaneous $Z_2^{f}$ symmetry breaking, and long range edge correlation in Kitaev chains: analytic solutions and density-matrix-renormalization-group study

Jian-Jian Miao, Hui-Ke Jin, Fu‐Chun Zhang +1 more

We study Kitaev model in one-dimension with open boundary condition by using exact analytic methods for non-interacting system in the half filled limit as well as in the symmetric case of $\Delta=t$, and by using density-matrix-renormalization-group method for interacting system with nearest neighbor repulsion. We show explicitly that the emergence of Majorana zero mode is associated with the spontaneous symmetry broken of fermion number parity operator $Z_2^f$, which is $+1$ for even number of fermions and $-1$ for odd number of fermions. Our results are in support of recent proposal by Klassen and Wen\cite{Klassen}, who proposed such spontaneous symmetry broken in systems with Majorana zero modes. 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.

•Aug 30, 2016•0 citations

Phase diagram of the interacting Majorana chain model

Armin Rahmani, Xiaoyu Zhu, Marcel Franz +1 more

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.

arXiv:1505.03966•Physical Review B•Dec 14, 2015•117 citations

Exact ground states and topological order in interacting Kitaev/Majorana chains

Hosho Katsura, Dirk Schuricht, Masahiro O. Takahashi

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.

arXiv:1507.04444•Physical Review B•Sep 18, 2015•158 citations

Topological Phase Diagram of an Interacting Kitaev Chain: Mean Field versus DMRG Study

Giovanni Nunziante, Alfonso Maiellaro, Claudio Guarcello +1 more

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.

Condensed Matter•Mar 20, 2024•0 citations

Robustness of Majorana edge modes and topological order: Exact results for the symmetric interacting Kitaev chain with disorder

Max McGinley, Johannes Knolle, Andreas Nunnenkamp

We investigate the robustness of Majorana edge modes under disorder and interactions. We exploit a recently found mapping of the interacting Kitaev chain in the symmetric region ($\ensuremath{\mu}=0, t=\mathrm{\ensuremath{\Delta}}$) to free fermions. Extending the exact solution to the disordered case allows us to calculate analytically the topological phase boundary for all interaction and disorder strengths, which has been thought to be only accessible numerically. We discover a regime in which moderate disorder in the interaction matrix elements enhances topological order well into the strongly interacting regime $U>t$. We also derive the explicit form of the many-body Majorana edge wave function, revealing how it is dressed by many-particle fluctuations from interactions. The qualitative features of our analytical results are valid beyond the fine-tuned integrable point, as expected from the robustness of topological order and as corroborated here by an exact diagonalization study of small systems.

arXiv:1706.10249•Physical review. B./Physical review. B•Dec 27, 2017•42 citations

Effect of Interaction on the Majorana Zero Modes in the Kitaev Chain at Half Filling

Zhidan Li, Qiang Han

The one-dimensional interacting Kitaev chain at half filling is studied. The symmetry of the Hamiltonian is examined by dual transformations, and various physical quantities as a function of the fermion-fermion interaction U are calculated systematically using the density matrix renormalization group method. A special value of interaction U p is revealed in the topological region of the phase diagram. We show that at U p the ground states are strictly two-fold degenerate even though the chain length is finite and the zero-energy peak due to the Majorana zero modes is maximally enhanced and exactly localized at the end sites. Here U p may be attractive or repulsive depending on other system parameters. We also give a qualitative understanding of the effect of interaction under the self-consistent mean field framework.

arXiv:1805.01632•Chinese Physics Letters•Apr 1, 2018•7 citations
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.
Majorana zero modes, spontaneous Z2fZ_2^{f}Z2f​ symmetry breaking, and long range edge correlation in Kitaev chains: analytic solutions and density-matrix-renormalization-group study
We study Kitaev model in one-dimension with open boundary condition by using exact analytic methods for non-interacting system in the half filled limit as well as in the symmetric case of Δ=t\Delta=tΔ=t, and by using density-matrix-renormalization-group method for interacting system with nearest neighbor repulsion. We show explicitly that the emergence of Majorana zero mode is associated with the spontaneous symmetry broken of fermion number parity operator Z2fZ_2^fZ2f​, which is +1+1+1 for even number of fermions and −1-1−1 for odd number of fermions. Our results are in support of recent proposal by Klassen and Wen\cite{Klassen}, who proposed such spontaneous symmetry broken in systems with Majorana zero modes. 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.
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 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.
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.
We investigate the robustness of Majorana edge modes under disorder and interactions. We exploit a recently found mapping of the interacting Kitaev chain in the symmetric region ($\ensuremath{\mu}=0, t=\mathrm{\ensuremath{\Delta}}$) to free fermions. Extending the exact solution to the disordered case allows us to calculate analytically the topological phase boundary for all interaction and disorder strengths, which has been thought to be only accessible numerically. We discover a regime in which moderate disorder in the interaction matrix elements enhances topological order well into the strongly interacting regime U>tU>tU>t. We also derive the explicit form of the many-body Majorana edge wave function, revealing how it is dressed by many-particle fluctuations from interactions. The qualitative features of our analytical results are valid beyond the fine-tuned integrable point, as expected from the robustness of topological order and as corroborated here by an exact diagonalization study of small systems.
The one-dimensional interacting Kitaev chain at half filling is studied. The symmetry of the Hamiltonian is examined by dual transformations, and various physical quantities as a function of the fermion-fermion interaction U are calculated systematically using the density matrix renormalization group method. A special value of interaction U p is revealed in the topological region of the phase diagram. We show that at U p the ground states are strictly two-fold degenerate even though the chain length is finite and the zero-energy peak due to the Majorana zero modes is maximally enhanced and exactly localized at the end sites. Here U p may be attractive or repulsive depending on other system parameters. We also give a qualitative understanding of the effect of interaction under the self-consistent mean field framework.