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:1903.08201 · Nuclear Physics B

Real fermion modes, impurity entropy, and nontrivial fixed points in the phase diagram of junctions of interacting quantum wires and topological superconductors

Domenico Giuliano, Ian K. Affleck

Published May 21, 2019• 10 citations
arXivHTMLPDF

Abstract

We discuss how to extend the impurity entropy to systems with boundary interactions depending on zero-mode real fermion operators (Majorana modes as well as Klein factors). As specific applications of our method, we consider a junction between N interacting quantum wires and a topological superconductor, as well as a Y-junction of three spinless interacting quantum wires. In addition we find a remarkable correspondence between the N=2 topological superconductor junction and the Y-junction. On one hand, this allows us to determine the range of the system parameters in which a stable phase of the N=2 junction is realized as a nontrivial, finite-coupling fixed point corresponding to the M-fixed point in the phase diagram of the Y-junction. On the other hand, it enables us to show the occurrence of a novel “planar” finite-coupling fixed point in the phase diagram of the Y-junction. Eventually, we discuss how to set the system's parameters to realize the correspondence.

DOI
10.1016/j.nuclphysb.2019.114645

Related papers

Impurity entropy of junctions of multiple quantum wires

Zheng Shi

Abstract We calculate the zero-temperature impurity entropy of a junction of multiple quantum wires of interacting spinless fermions. Starting from a given single-particle S-matrix representing a fixed point of the renormalization group (RG) flows, we carry out fermionic perturbation theory in the bulk interactions, with the perturbation series summed in the random phase approximation (RPA). The results agree completely with boundary conformal field theory (BCFT) predictions of the ground state degeneracy, and also with known RG flows through the g -theorem.

arXiv:1602.00068•Journal of Statistical Mechanics Theory and Experiment•Jun 1, 2016•3 citations

Quantum Criticality in Topological Insulators and Superconductors: Emergence of Strongly Coupled Majoranas and Supersymmetry

Tarun Grover, Ashvin Vishwanath

We study symmetry breaking quantum phase transitions in topological insulators and superconductors where the single electron gap remains open in the bulk. Specifically, we consider spontaneous breaking of the symmetry that protects the gapless boundary modes, so that in the ordered phase these modes are gapped. Here we determine the fate of the topological boundary modes right at the transition where they are coupled to the strongly fluctuating order parameter field. Using a combination of exact solutions and renormalization group techniques, we find that the surface fermionic modes either decouple from the bulk fluctuations, or flow to a strongly coupled fixed point which remains gapless. In addition, we study transitions where the critical fluctuations are confined only to the surface and find that in several cases the critical point is naturally supersymmetric. This allows a determination of critical exponents and points to an underlying connection between band topology and supersymmetry. Finally, we study the fate of gapless Majorana modes localized on point and line defects in topological superconductors at bulk criticality, which is analogous to a quantum impurity problem. Again, an interplay of topology and strong correlations causes these modes to remain gapless but in a strongly coupled state. Experimental candidates for realizing these phenomena are discussed.

arXiv:1206.1332•Jun 6, 2012•24 citations

Junctions of multiple quantum wires with different Luttinger parameters

Chang-Yu Hou, Armin Rahmani, Adrian E. Feiguin +1 more

Within the framework of boundary conformal field theory, we evaluate the conductance of stable fixed points of junctions of two and three quantum wires with different Luttinger parameters. For two wires, the physical properties are governed by a single effective Luttinger parameter for each of the charge and spin sectors. We present numerical density-matrix-renormalization-group calculations of the conductance of a junction of two chains of interacting spinless fermions with different interaction strengths, obtained using a recently developed method [Phys. Rev. Lett. 105, 226803 (2010)]. The numerical results show very good agreement with the analytical predictions. For three spinless wires (i.e., a Y junction) we analytically determine the full phase diagram and compute all fixed-point conductances as a function of the three Luttinger parameters.

arXiv:1205.2125•Physical Review B•Aug 22, 2012•33 citations

Phase diagram and entanglement of two interacting topological Kitaev chains

Loïc Herviou, Christophe Mora, Karyn Le Hur

A superconducting wire described by a $p$-wave pairing and a Kitaev Hamiltonian exhibits Majorana fermions at its edges and is topologically protected by symmetry. We consider two Kitaev wires (chains) coupled by a Coulomb-type interaction and study the complete phase diagram using analytical and numerical techniques. A topological superconducting phase with four Majorana fermions occurs until moderate interactions between chains. For large interactions, both repulsive and attractive, by analogy with the Hubbard model, we identify Mott phases with Ising-type magnetic order. For repulsive interactions, the Ising antiferromagnetic order favors the occurrence of orbital currents spontaneously breaking time-reversal symmetry. By strongly varying the chemical potentials of the two chains, quantum phase transitions towards fully polarized (empty or full) fermionic chains occur. In the Kitaev model, the quantum critical point separating the topological superconducting phase and the polarized phase belongs to the universality class of the critical Ising model in two dimensions. When increasing the Coulomb interaction between chains, then we identify an additional phase corresponding to two critical Ising theories (or two chains of Majorana fermions). We confirm the existence of such a phase from exact mappings and from the concept of bipartite fluctuations. We show the existence of negative logarithmic corrections in the bipartite fluctuations, as a reminiscence of the quantum critical point in the Kitaev model. Other entanglement probes such as bipartite entropy and entanglement spectrum are also used to characterize the phase diagram. The limit of large interactions can be reached in an equivalent setup of ultracold atoms and Josephson junctions.

arXiv:1601.02998•Physical review. B./Physical review. B•Apr 27, 2016•21 citations

Exact zero modes in interacting Majorana X - and Y -junctions

Bowy La Rivière, Rik Mulder, Natalia Chepiga

Abstract We report the emergence of exact zero modes (EZMs) in junctions of two, three and four short interacting Majorana wires, equivalent to a chain with an impurity bond, Y - and X - junctions respectively. These EZMs arise from incommensurate short-range correlations and manifest as exact level crossings between in-gap states upon continuously tuning the interaction strength. In a junction of two chains the EZMs and parity switching appear as soon as the coupling across a junction is positive. This conclusion has a broad implication including the existence of EZM in quench- disordered systems. Remarkably, for junctions with multiple chains the in-gap states group up into sets of parity pairs–pairs of quasi-degenerate states with opposite parity attributed to the effective interaction of the outer edges of the junction. In the Y -junction with four in-gap states, we report EZMs not only within each parity pair but also between them. We demonstrate that this is attributed to an additional Majorana fermion localized at the center of the junction and protected by fermion parity symmetry. Therefore, coupling between the Majorana fermions at the outer edges of the junction is mediated by the central one. We argue that this is a generic feature of junctions with an odd number of arms. Surprisingly, in the X -junction we identify eight in-gap states arising from two Majorana degrees of freedom localized at the junction center. In contrast to the Y -junction, however, we show that the emergence of Majorana modes at the center of the X -junction is not topologically protected. As a result, the system parameters can be tuned such that the Majorana modes are expelled from the junction center and remain localized exclusively at the outer edges of the system.

arXiv:2506.10898•New Journal of Physics•Sep 11, 2026•1 citation

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
We discuss how to extend the impurity entropy to systems with boundary interactions depending on zero-mode real fermion operators (Majorana modes as well as Klein factors). As specific applications of our method, we consider a junction between N interacting quantum wires and a topological superconductor, as well as a Y-junction of three spinless interacting quantum wires. In addition we find a remarkable correspondence between the N=2 topological superconductor junction and the Y-junction. On one hand, this allows us to determine the range of the system parameters in which a stable phase of the N=2 junction is realized as a nontrivial, finite-coupling fixed point corresponding to the M-fixed point in the phase diagram of the Y-junction. On the other hand, it enables us to show the occurrence of a novel “planar” finite-coupling fixed point in the phase diagram of the Y-junction. Eventually, we discuss how to set the system's parameters to realize the correspondence.
Abstract We calculate the zero-temperature impurity entropy of a junction of multiple quantum wires of interacting spinless fermions. Starting from a given single-particle S-matrix representing a fixed point of the renormalization group (RG) flows, we carry out fermionic perturbation theory in the bulk interactions, with the perturbation series summed in the random phase approximation (RPA). The results agree completely with boundary conformal field theory (BCFT) predictions of the ground state degeneracy, and also with known RG flows through the g -theorem.
We study symmetry breaking quantum phase transitions in topological insulators and superconductors where the single electron gap remains open in the bulk. Specifically, we consider spontaneous breaking of the symmetry that protects the gapless boundary modes, so that in the ordered phase these modes are gapped. Here we determine the fate of the topological boundary modes right at the transition where they are coupled to the strongly fluctuating order parameter field. Using a combination of exact solutions and renormalization group techniques, we find that the surface fermionic modes either decouple from the bulk fluctuations, or flow to a strongly coupled fixed point which remains gapless. In addition, we study transitions where the critical fluctuations are confined only to the surface and find that in several cases the critical point is naturally supersymmetric. This allows a determination of critical exponents and points to an underlying connection between band topology and supersymmetry. Finally, we study the fate of gapless Majorana modes localized on point and line defects in topological superconductors at bulk criticality, which is analogous to a quantum impurity problem. Again, an interplay of topology and strong correlations causes these modes to remain gapless but in a strongly coupled state. Experimental candidates for realizing these phenomena are discussed.
Within the framework of boundary conformal field theory, we evaluate the conductance of stable fixed points of junctions of two and three quantum wires with different Luttinger parameters. For two wires, the physical properties are governed by a single effective Luttinger parameter for each of the charge and spin sectors. We present numerical density-matrix-renormalization-group calculations of the conductance of a junction of two chains of interacting spinless fermions with different interaction strengths, obtained using a recently developed method [Phys. Rev. Lett. 105, 226803 (2010)]. The numerical results show very good agreement with the analytical predictions. For three spinless wires (i.e., a Y junction) we analytically determine the full phase diagram and compute all fixed-point conductances as a function of the three Luttinger parameters.
A superconducting wire described by a ppp-wave pairing and a Kitaev Hamiltonian exhibits Majorana fermions at its edges and is topologically protected by symmetry. We consider two Kitaev wires (chains) coupled by a Coulomb-type interaction and study the complete phase diagram using analytical and numerical techniques. A topological superconducting phase with four Majorana fermions occurs until moderate interactions between chains. For large interactions, both repulsive and attractive, by analogy with the Hubbard model, we identify Mott phases with Ising-type magnetic order. For repulsive interactions, the Ising antiferromagnetic order favors the occurrence of orbital currents spontaneously breaking time-reversal symmetry. By strongly varying the chemical potentials of the two chains, quantum phase transitions towards fully polarized (empty or full) fermionic chains occur. In the Kitaev model, the quantum critical point separating the topological superconducting phase and the polarized phase belongs to the universality class of the critical Ising model in two dimensions. When increasing the Coulomb interaction between chains, then we identify an additional phase corresponding to two critical Ising theories (or two chains of Majorana fermions). We confirm the existence of such a phase from exact mappings and from the concept of bipartite fluctuations. We show the existence of negative logarithmic corrections in the bipartite fluctuations, as a reminiscence of the quantum critical point in the Kitaev model. Other entanglement probes such as bipartite entropy and entanglement spectrum are also used to characterize the phase diagram. The limit of large interactions can be reached in an equivalent setup of ultracold atoms and Josephson junctions.
Abstract We report the emergence of exact zero modes (EZMs) in junctions of two, three and four short interacting Majorana wires, equivalent to a chain with an impurity bond, Y - and X - junctions respectively. These EZMs arise from incommensurate short-range correlations and manifest as exact level crossings between in-gap states upon continuously tuning the interaction strength. In a junction of two chains the EZMs and parity switching appear as soon as the coupling across a junction is positive. This conclusion has a broad implication including the existence of EZM in quench- disordered systems. Remarkably, for junctions with multiple chains the in-gap states group up into sets of parity pairs–pairs of quasi-degenerate states with opposite parity attributed to the effective interaction of the outer edges of the junction. In the Y -junction with four in-gap states, we report EZMs not only within each parity pair but also between them. We demonstrate that this is attributed to an additional Majorana fermion localized at the center of the junction and protected by fermion parity symmetry. Therefore, coupling between the Majorana fermions at the outer edges of the junction is mediated by the central one. We argue that this is a generic feature of junctions with an odd number of arms. Surprisingly, in the X -junction we identify eight in-gap states arising from two Majorana degrees of freedom localized at the junction center. In contrast to the Y -junction, however, we show that the emergence of Majorana modes at the center of the X -junction is not topologically protected. As a result, the system parameters can be tuned such that the Majorana modes are expelled from the junction center and remain localized exclusively at the outer edges of the system.
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.