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

Low Temperature Physics

Ground state of the biaxial spin-1/2 open chain

Andrei A. Zvyagin

Published May 1, 2022• 1 citation
Publisher

Abstract

The ground state behavior of the biaxial spin-1/2 chain with free open edges is studied. Using the exact Bethe ansatz solution we show that there can exist boundary bound states for many finite values of the exchange coupling constants. The non-trivial interaction between spins produces charging of the vacua of the model and boundary bound states. Our theory also describes the behavior of the spinless fermion chain with pairing (the Kitaev chain) and an interaction between fermions at neighboring sites for free open boundaries. Therefore, the simple case of noninteracting fermions simplest boundary states are Majorana edge modes.

DOI
10.1063/10.0010202

Related papers

Charging of Majorana edge modes caused by interaction: Exact results

Andrei A. Zvyagin

The ground-state behavior of a spinless fermion chain with pairing (the Kitaev chain) and an interaction between fermions at neighboring sites is studied for free open boundaries. Using the exact quantum solution, it has been shown that there can exist boundary bound states for many values of the interaction. The interaction produces charging of the vacua of the model and charging of the boundary bound states. The theory also describes the behavior of an XYZ spin-1/2 chain with free open edges.

Physical review. B./Physical review. B•Mar 9, 2022•6 citations

Majorana zero modes in the interacting fermion chain without pairing

Andrei A. Zvyagin

The ground state behavior of the spinless fermion chain with an interaction between fermions at neighboring sites is studied for free open boundaries. For the strong enough repulsion boundary gapless states (bound state of Majorana operators from opposite sites of the chain) can exist inside the gap for bulk excitations, i.e., in the topological insulator regime. We propose to use those Majorana zero modes as topological qubits, similar to the ones in one-dimensional topological superconductors. Possible physical realizations of the considered model are discussed.

Low Temperature Physics•May 1, 2021•6 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

Expectation value of the edge Majorana fermion in an interacting fermion chain

Andrei A. Zvyagin

The expectation value of the edge Majorana operator is calculated for the one-dimensional model of spinless fermions with nearest-neighbor interactions and with open boundary conditions. The consideration is performed for the regime of the topological insulator (gapped bulk excitations and edge modes inside the gap), and for the normal metal regime (gapless bulk excitations). We show that the expectation value of the edge Majorana operator can be formed by two contributions, from the bulk states, and from the localized edge modes.

Physical review. B./Physical review. B•May 19, 2021•3 citations

Open XXZ chain and boundary modes at zero temperature

Sebastian Grijalva, Jacopo De Nardis, V. Terras

We study the open XXZ spin chain in the anti-ferromagnetic regime and for generic longitudinal magnetic fields at the two boundaries. We discuss the ground state via the Bethe ansatz and we show that, for a chain of even length L and in a regime where both boundary magnetic fields are equal and bounded by a critical field, the spectrum is gapped and the ground state is doubly degenerate up to exponentially small corrections in L. We connect this degeneracy to the presence of a boundary root, namely an excitation localized at one of the two boundaries. We compute the local magnetization at the left edge of the chain and we show that, due to the existence of a boundary root, this depends also on the value of the field at the opposite edge, even in the half-infinite chain limit. Moreover we give an exact expression for the large time limit of the spin autocorrelation at the boundary, which we explicitly compute in terms of the form factor between the two quasi-degenerate ground states. This, as we show, turns out to be equal to the contribution of the boundary root to the local magnetization. We finally discuss the case of chains of odd length.

arXiv:1901.10932•SciPost Physics•Aug 20, 2019•28 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
The ground state behavior of the biaxial spin-1/2 chain with free open edges is studied. Using the exact Bethe ansatz solution we show that there can exist boundary bound states for many finite values of the exchange coupling constants. The non-trivial interaction between spins produces charging of the vacua of the model and boundary bound states. Our theory also describes the behavior of the spinless fermion chain with pairing (the Kitaev chain) and an interaction between fermions at neighboring sites for free open boundaries. Therefore, the simple case of noninteracting fermions simplest boundary states are Majorana edge modes.
The ground state behavior of the spinless fermion chain with an interaction between fermions at neighboring sites is studied for free open boundaries. For the strong enough repulsion boundary gapless states (bound state of Majorana operators from opposite sites of the chain) can exist inside the gap for bulk excitations, i.e., in the topological insulator regime. We propose to use those Majorana zero modes as topological qubits, similar to the ones in one-dimensional topological superconductors. Possible physical realizations of the considered model are discussed.
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 expectation value of the edge Majorana operator is calculated for the one-dimensional model of spinless fermions with nearest-neighbor interactions and with open boundary conditions. The consideration is performed for the regime of the topological insulator (gapped bulk excitations and edge modes inside the gap), and for the normal metal regime (gapless bulk excitations). We show that the expectation value of the edge Majorana operator can be formed by two contributions, from the bulk states, and from the localized edge modes.
We study the open XXZ spin chain in the anti-ferromagnetic regime and for generic longitudinal magnetic fields at the two boundaries. We discuss the ground state via the Bethe ansatz and we show that, for a chain of even length L and in a regime where both boundary magnetic fields are equal and bounded by a critical field, the spectrum is gapped and the ground state is doubly degenerate up to exponentially small corrections in L. We connect this degeneracy to the presence of a boundary root, namely an excitation localized at one of the two boundaries. We compute the local magnetization at the left edge of the chain and we show that, due to the existence of a boundary root, this depends also on the value of the field at the opposite edge, even in the half-infinite chain limit. Moreover we give an exact expression for the large time limit of the spin autocorrelation at the boundary, which we explicitly compute in terms of the form factor between the two quasi-degenerate ground states. This, as we show, turns out to be equal to the contribution of the boundary root to the local magnetization. We finally discuss the case of chains of odd length.
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.