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:1910.11221 · Physical review. B./Physical review. B

Exact boundary modes in an interacting quantum wire

Colin Rylands

Published Feb 21, 2020• 14 citations
arXivHTMLPDF

Abstract

The boundary modes of one-dimensional quantum systems can play host to a variety of remarkable phenomena. They can be used to describe the physics of impurities in higher-dimensional systems, such as the ubiquitous Kondo effect, or can support Majorana bound states, which play a crucial role in the realm of quantum computation. In this work we examine the boundary modes in an interacting quantum wire with a proximity-induced pairing term. We solve the system exactly using the Bethe ansatz and show that for certain boundary conditions the spectrum contains bound states localized about either edge. The model is shown to exhibit a first-order phase transition as a function of the interaction strength such that for attractive interactions the ground state has bound states at both ends of the wire, while for repulsive interactions they are absent. In addition we see that the bound-state energy lies within the gap for all values of the interaction strength but undergoes a sharp avoided level crossing for sufficiently strong interaction, thereby preventing its decay. This avoided crossing is shown to occur as a consequence of an exact self-duality which is present in the model.

DOI
10.1103/physrevb.101.085133

Related papers

Ansatz for Majorana modes for a quantum walk in a finite wire

Ho Tat Lam, Yue Yu, Kwok Yip Szeto

The topological phases in a one-dimensional quantum walk can be classified by the coin parameters. By solving for the general exact solutions of bound states in a one-dimensional quantum walk with boundaries specified by different coin parameters, we show that these bound states are Majorana modes with quasienergy $E=0,\ensuremath{\pi}$. These modes are qualitatively different for different boundary conditions used. For systems with two boundaries, such as finite wires, we find the same exact solution as the single boundary system if the boundary condition is antisymmetric. For a two-boundary system with symmetric boundary conditions in a finite wire, the interaction energy between two Majorana bound states can be computed as a function of the length of the wire and the coin parameter in the wire. Suggestions of observing these modes are provided.

arXiv:1508.07528•Physical Review A•May 16, 2016•8 citations

Quantum impurities in interacting environments

Colin Rylands

The quantum physics of one dimensional systems exhibits many remarkable and exotic physical phenomena. These are the result of the strong correlations which exist in such systems due to the reduced dimensionality and in many cases can be described by Luttinger Liquid theory. Likewise quantum impurity systems, where a bath of particles couples to the same impurity, exhibit well known non perturbative phenomena like the Kondo effect. In this thesis we study the physics of quantum impurities in interacting environments - the intersection of these two areas. Such Luttinger-impurity systems can be realized experimentally in a range of different settings from quantum wires and carbon nano tubes to edges of fractional quantum hall systems and cold atom gases. We show that many of the models used to describe these experiments are integrable and can be studied by means of the Bethe Ansatz. This powerful and exact method provides us with the exact eigenstates and energy levels of the model Hamiltonian. The models considered include two which describe a Lutttinger liquid coupled to a back scattering impurity, the Kane-Fisher and weak tunnelling model as well as Luttinger liquids coupled a quantum dot in various geometries: at the boundary, sidecoupled and embedded. To incorporate both the backscattering nature of the impurities and the interacting bulk particles a new formulation of the Bethe Ansatz is developed and implemented. The eigenstates and spectra of the models are found and used to investigate their ground state and thermodynamic properties.

Bulletin of the American Physical Society•Jan 1, 2018•0 citations

Emergent boundary supersymmetry in a one-dimensional superconductor

Parameshwar R. Pasnoori, P. Azaria, Colin Rylands +1 more

The interplay between bulk properties and boundary conditions in one-dimensional quantum systems, gives rise to many intriguing phenomena. These include the emergence of zero energy modes which are of significant interest to a variety of fields. In this work we investigate the presence of such zero modes in cases where the boundary conditions are dynamical and arise due to the coupling to some quantum degrees of freedom. In particular, we study a one-dimensional spin-singlet superconductor, modeled by the Gross-Neveu field theory, coupled to spin $\frac{1}{2}$ magnetic impurities at its boundaries via a spin-exchange interaction. We solve the model exactly for arbitrary values of the bulk and the impurity coupling strengths using nested coordinate Bethe ansatz and show that the system exhibits a rich boundary phase structure. For a range of couplings, the low energy degrees of freedom form irreducible representations of the supersymmetric $spl(2,1)\otimes spl(2,1)$ algebra which become degenerate at a specific point, indicating the emergence of supersymmetry in the low energy boundary degrees of freedom. We show that at the supersymmetric point there exist exact zero energy modes that map one ground state with the other. We express these in terms of the generators of the algebra.

arXiv:2505.24777•Physical review. B./Physical review. B•Jan 9, 2026•0 citations

Spectrum of boundary states in the open Hubbard chain

Gerald Bedürftig, Holger Frahm

We use the Bethe ansatz solution for the one-dimensional Hubbard model with open boundary conditions and applied boundary fields to study the spectrum of bound states at the boundary. Depending on the strength of the boundary potentials, one finds that the true ground state contains a single charge or, for boundary potentials comparable with the Hubbard interaction, a pair of electrons in a bound state. If these are left unoccupied one finds holon and spinon bound states. We compute the finite size corrections to the low-lying energies in this system and use the predictions of boundary conformal field theory to study the exponents related to the orthogonality catastrophe.

arXiv:cond-mat/9702227•Journal of Physics A Mathematical and General•Jun 21, 1997•42 citations

Thermodynamics of the topological Kondo model

Francesco Buccheri, H. Babujian, V. E. Korepin +2 more

Using the thermodynamic Bethe ansatz, we investigate the topological Kondo model, which describes a set of one-dimensional external wires, pertinently coupled to a central region hosting a set of Majorana bound states. After a short review of the Bethe ansatz solution, we study the system at finite temperature and derive its free energy for arbitrary (even and odd) number of external wires. We then analyse the ground state energy as a function of the number of external wires and of their couplings to the Majorana bound states. Then, we compute, both for small and large temperatures, the entropy of the Majorana degrees of freedom localized within the central region and connected to the external wires. Our exact computation of the impurity entropy provides evidence of the importance of fermion parity symmetry in the realization of the topological Kondo model. Finally, we also obtain the low-temperature behaviour of the specific heat of the Majorana bound states, which provides a signature of the non-Fermi-liquid nature of the strongly coupled fixed point.

arXiv:1412.5292•Nuclear Physics B•Apr 21, 2015•26 citations

Corner Majorana states in semi-Dirac materials

M. García Olmos, Y. Baba, R. A. Molina +1 more

Proximity-induced superconductivity in low-dimensional systems offers a powerful pathway to engineer topological superconducting phases in, otherwise, non-superconducting systems. These exotic phases are of fundamental and technological interest due to the presence of robust zero-energy modes, the Majorana bound states. In this work, we propose a theoretical framework to realize Majorana bound states from the edge states of a two-dimensional semi-Dirac system. This anisotropic system, under specific conditions, can host non-chiral edge states that propagate only along particular edges, effectively forming separated one-dimensional channels. We show that the interplay between Rashba spin-orbit coupling and a Zeeman field on this setup provides the right conditions to get an effective p-wave pairing between the edge states by proximity with a s-wave superconductor. In finite geometries, each edge can independently undergo a topological phase transition into a one-dimensional topological superconductor and give rise to four zero-energy modes localized at the strip corners. At low energies, the edge states subspace admits a description in terms of coupled Kitaev chains, providing a clear picture of the origin, robustness, and tunability of the corner Majorana modes. Our results establish semi-Dirac materials as a natural platform for realizing Majorana modes in two dimensions without relying on engineered nanostructures, vortices, or crystalline higher-order topology.

cond-mat.mes-hallCross-listedarXiv:2604.22553•Apr 27, 2026
The boundary modes of one-dimensional quantum systems can play host to a variety of remarkable phenomena. They can be used to describe the physics of impurities in higher-dimensional systems, such as the ubiquitous Kondo effect, or can support Majorana bound states, which play a crucial role in the realm of quantum computation. In this work we examine the boundary modes in an interacting quantum wire with a proximity-induced pairing term. We solve the system exactly using the Bethe ansatz and show that for certain boundary conditions the spectrum contains bound states localized about either edge. The model is shown to exhibit a first-order phase transition as a function of the interaction strength such that for attractive interactions the ground state has bound states at both ends of the wire, while for repulsive interactions they are absent. In addition we see that the bound-state energy lies within the gap for all values of the interaction strength but undergoes a sharp avoided level crossing for sufficiently strong interaction, thereby preventing its decay. This avoided crossing is shown to occur as a consequence of an exact self-duality which is present in the model.
The topological phases in a one-dimensional quantum walk can be classified by the coin parameters. By solving for the general exact solutions of bound states in a one-dimensional quantum walk with boundaries specified by different coin parameters, we show that these bound states are Majorana modes with quasienergy $E=0,\ensuremath{\pi}$. These modes are qualitatively different for different boundary conditions used. For systems with two boundaries, such as finite wires, we find the same exact solution as the single boundary system if the boundary condition is antisymmetric. For a two-boundary system with symmetric boundary conditions in a finite wire, the interaction energy between two Majorana bound states can be computed as a function of the length of the wire and the coin parameter in the wire. Suggestions of observing these modes are provided.
The quantum physics of one dimensional systems exhibits many remarkable and exotic physical phenomena. These are the result of the strong correlations which exist in such systems due to the reduced dimensionality and in many cases can be described by Luttinger Liquid theory. Likewise quantum impurity systems, where a bath of particles couples to the same impurity, exhibit well known non perturbative phenomena like the Kondo effect. In this thesis we study the physics of quantum impurities in interacting environments - the intersection of these two areas. Such Luttinger-impurity systems can be realized experimentally in a range of different settings from quantum wires and carbon nano tubes to edges of fractional quantum hall systems and cold atom gases. We show that many of the models used to describe these experiments are integrable and can be studied by means of the Bethe Ansatz. This powerful and exact method provides us with the exact eigenstates and energy levels of the model Hamiltonian. The models considered include two which describe a Lutttinger liquid coupled to a back scattering impurity, the Kane-Fisher and weak tunnelling model as well as Luttinger liquids coupled a quantum dot in various geometries: at the boundary, sidecoupled and embedded. To incorporate both the backscattering nature of the impurities and the interacting bulk particles a new formulation of the Bethe Ansatz is developed and implemented. The eigenstates and spectra of the models are found and used to investigate their ground state and thermodynamic properties.
The interplay between bulk properties and boundary conditions in one-dimensional quantum systems, gives rise to many intriguing phenomena. These include the emergence of zero energy modes which are of significant interest to a variety of fields. In this work we investigate the presence of such zero modes in cases where the boundary conditions are dynamical and arise due to the coupling to some quantum degrees of freedom. In particular, we study a one-dimensional spin-singlet superconductor, modeled by the Gross-Neveu field theory, coupled to spin 12\frac{1}{2}21​ magnetic impurities at its boundaries via a spin-exchange interaction. We solve the model exactly for arbitrary values of the bulk and the impurity coupling strengths using nested coordinate Bethe ansatz and show that the system exhibits a rich boundary phase structure. For a range of couplings, the low energy degrees of freedom form irreducible representations of the supersymmetric spl(2,1)⊗spl(2,1)spl(2,1)\otimes spl(2,1)spl(2,1)⊗spl(2,1) algebra which become degenerate at a specific point, indicating the emergence of supersymmetry in the low energy boundary degrees of freedom. We show that at the supersymmetric point there exist exact zero energy modes that map one ground state with the other. We express these in terms of the generators of the algebra.
We use the Bethe ansatz solution for the one-dimensional Hubbard model with open boundary conditions and applied boundary fields to study the spectrum of bound states at the boundary. Depending on the strength of the boundary potentials, one finds that the true ground state contains a single charge or, for boundary potentials comparable with the Hubbard interaction, a pair of electrons in a bound state. If these are left unoccupied one finds holon and spinon bound states. We compute the finite size corrections to the low-lying energies in this system and use the predictions of boundary conformal field theory to study the exponents related to the orthogonality catastrophe.
Using the thermodynamic Bethe ansatz, we investigate the topological Kondo model, which describes a set of one-dimensional external wires, pertinently coupled to a central region hosting a set of Majorana bound states. After a short review of the Bethe ansatz solution, we study the system at finite temperature and derive its free energy for arbitrary (even and odd) number of external wires. We then analyse the ground state energy as a function of the number of external wires and of their couplings to the Majorana bound states. Then, we compute, both for small and large temperatures, the entropy of the Majorana degrees of freedom localized within the central region and connected to the external wires. Our exact computation of the impurity entropy provides evidence of the importance of fermion parity symmetry in the realization of the topological Kondo model. Finally, we also obtain the low-temperature behaviour of the specific heat of the Majorana bound states, which provides a signature of the non-Fermi-liquid nature of the strongly coupled fixed point.
Proximity-induced superconductivity in low-dimensional systems offers a powerful pathway to engineer topological superconducting phases in, otherwise, non-superconducting systems. These exotic phases are of fundamental and technological interest due to the presence of robust zero-energy modes, the Majorana bound states. In this work, we propose a theoretical framework to realize Majorana bound states from the edge states of a two-dimensional semi-Dirac system. This anisotropic system, under specific conditions, can host non-chiral edge states that propagate only along particular edges, effectively forming separated one-dimensional channels. We show that the interplay between Rashba spin-orbit coupling and a Zeeman field on this setup provides the right conditions to get an effective p-wave pairing between the edge states by proximity with a s-wave superconductor. In finite geometries, each edge can independently undergo a topological phase transition into a one-dimensional topological superconductor and give rise to four zero-energy modes localized at the strip corners. At low energies, the edge states subspace admits a description in terms of coupled Kitaev chains, providing a clear picture of the origin, robustness, and tunability of the corner Majorana modes. Our results establish semi-Dirac materials as a natural platform for realizing Majorana modes in two dimensions without relying on engineered nanostructures, vortices, or crystalline higher-order topology.