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:2311.01880 · Physical review. B./Physical review. B

Topological phases of strongly interacting time-reversal invariant topological superconducting chains under a magnetic field

Leandro M. Chinellato, C. J. Gazza, Alejandro M. Lobos, Armando A. Aligia

Published Feb 8, 2024• 6 citations
arXivHTMLPDF

Abstract

Using the density-matrix renormalization group, we determine the different topological phases and low-energy excitations of a time-reversal invariant topological superconducting (TRITOPS) wire with extended $s$-wave superconductivity, Rashba spin-orbit coupling (SOC) and on-site repulsion $U$, under an externally applied Zeeman field $J$. For the case in which $J$ is perpendicular to the SOC, the model describes a chain of Shiba impurities on top of a superconductor with extended superconductor pairing. We identify the different topological phases of the model at temperature $T=0$, and in particular we study the stability of the TRITOPS phase against the Zeeman field $J$ and the chemical potential $\ensuremath{\mu}$, for different values of $U$. In the case where the magnetic field $J$ is perpendicular to the SOC axis, the pair of Kramers degenerate Majorana zero modes at the edges of the system that exist for $J=0$, remain degenerate until a critical value of the magnetic field is reached. For $J$ parallel to the SOC and up to moderate values of $U$, the fractional spin projection $\ensuremath{\langle}{S}_{y}\ensuremath{\rangle}=1/4$ at the ends, found for noninteracting wires at $U=0$, is recovered. In addition, the analytic expression that relates $\ensuremath{\langle}{S}_{y}\ensuremath{\rangle}$ with $J$ for finite noninteracting chains is shown to be universal up to moderate values of $U$.

DOI
10.1103/physrevb.109.064503

Related papers

Exact solution of a time-reversal-invariant topological superconducting wire

Armando A. Aligia, Alberto Camjayi

We consider a model proposed before for a time-reversal-invariant topological superconductor (TRITOPS) which contains a hopping term $t$, a chemical potential $\mu$, an extended $s$-wave pairing $\Delta$ and spin-orbit coupling $\lambda$. We show that for $|\Delta|=|\lambda|$, $\mu=t=0$, the model can be solved exactly defining new fermion operators involving nearest-neighbor sites. The many-body ground state is four-fold degenerate due to the existence of two zero-energy modes localized exactly at the first and the last site of the chain. These four states show entanglement in the sense that creating or annihilating a zero-energy mode at the first site is proportional to a similar operation at the last site. By continuity, this property should persist for general parameters. Using these results we correct some statements related with the so called time-reversal anomaly. Addition of a small hopping term for a chain with an even number of sites breaks the degeneracy and the ground state becomes unique with an even number of particles. We also consider a small magnetic field applied to one end of the chain. We compare the many-body excitation energies and spin projection along the spin-orbit direction for both ends of the chains with numerical results %for a small chain obtaining good agreement.

•May 10, 2019•0 citations

Entangled end states with fractionalized spin projection in a time-reversal-invariant topological superconducting wire

Armando A. Aligia, Liliana Arrachea

We study the ground state and low-energy subgap excitations of a finite wire of a time-reversal-invariant topological superconductor (TRITOPS) with spin-orbit coupling. We solve the problem analytically for a long chain of a specific one-dimensional lattice model in the electron-hole symmetric configuration and numerically for other cases of the same model. We present results for the spin density of excitations in long chains with an odd number of particles. The total spin projection along the axis of the spin-orbit coupling ${S}_{z}=\ifmmode\pm\else\textpm\fi{}1/2$ is distributed with fractions $\ifmmode\pm\else\textpm\fi{}1/4$ localized at both ends and shows even-odd alternation along the sites of the chain. We calculate the localization length of these excitations and find that it can be well approximated by a simple analytical expression. We show that the energy $E$ of the lowest subgap excitations of the finite chain defines tunneling and entanglement between end states. We discuss the effect of a Zeeman coupling ${\mathrm{\ensuremath{\Delta}}}_{Z}$ on one of the ends of the chain only. For ${\mathrm{\ensuremath{\Delta}}}_{Z}<E$, the energy difference of excitations with opposite spin orientation is ${\mathrm{\ensuremath{\Delta}}}_{Z}/2$, consistent with a spin projection $\ifmmode\pm\else\textpm\fi{}1/4$. We argue that these physical features are not model dependent and can be experimentally observed in TRITOPS wires under appropriate conditions.

arXiv:1806.06104•Physical review. B./Physical review. B•Nov 20, 2018•22 citations

Exact analytical solution of a time-reversal-invariant topological superconducting wire

Armando A. Aligia, Alberto Camjayi

We consider a model proposed before for a time-reversal-invariant topological superconductor which contains a hopping term $t$, a chemical potential $\ensuremath{\mu}$, an extended $s$-wave pairing $\mathrm{\ensuremath{\Delta}}$, and spin-orbit coupling $\ensuremath{\lambda}$. We show that for $|\mathrm{\ensuremath{\Delta}}|=|\ensuremath{\lambda}|$, $\ensuremath{\mu}=t=0$, the model has an exact analytical solution defining new fermion operators involving nearest-neighbor sites. The many-body ground state is fourfold degenerate due to the existence of two zero-energy modes localized exactly at the first and last sites of the chain. These four states show entanglement in the sense that creating or annihilating a zero-energy mode at the first site is proportional to a similar operation at the last site. By continuity, this property should persist for general parameters. Using these results, we discuss some statements related to the so-called time-reversal anomaly. The addition of a small hopping term $t$ for a chain with an even number of sites breaks the degeneracy, and the ground state becomes unique with an even number of particles. We also consider a small magnetic field $B$ applied to one end of the chain. We compare the many-body excitation energies and spin projection along the spin-orbit direction for both ends of the chains obtained treating $t$ and $B$ as small perturbations with numerical results in a short chain, obtaining good agreement.

arXiv:1905.04255•Physical review. B./Physical review. B•Sep 10, 2019•5 citations

Interaction effects in topological superconducting wires supporting Majorana fermions

Edwin Miles Stoudenmire, Jason Alicea, Oleg A. Starykh +1 more

Among the broad spectrum of systems predicted to exhibit topological superconductivity and Majorana fermions, one-dimensional wires with strong spin-orbit coupling provide one of the most promising experimental candidates. Here we investigate the fate of the topological superconducting phase in such wires when repulsive interactions are present. Using a combination of density matrix renormalization group, bosonization, and Hartree--Fock techniques, we demonstrate that while interactions degrade the bulk gap---consistent with recent results of Gangadharaiah et al.---they also greatly expand the parameter range over which the topological phase arises. In particular, we show that with interactions this phase can be accessed over a broader chemical potential window, thereby leading to greater immunity against disorder-induced chemical potential fluctuations in the wire. We also suggest that in certain wires strong interactions may allow Majorana fermions to be generated without requiring a magnetic field.

arXiv:1104.5493•Physical Review B•Jul 14, 2011•279 citations

Evolution of Majorona zero-energy edge states in a $T^2 = -1$ symmetry protected 1D topological superconductor with dominant spin-orbit coupling

Alestin Mawrie

We consider a 1D topological superconductor (TSC) constructed by coupling a pair of Kitaev's Majorana chains with opposite spin configurations. Such a 1D lattice model is known to be protected by a $T^2 = -1$ time-reversal symmetry. Furthermore, we consider a modeled Rashba spin-orbit coupling on such a system of $T^2=-1$ time-reversal symmetric TSC. The Rashba spin-orbit coupling together with the chemical potential engineered the phase transitions of the edge states in the system and consequently the number of Majorona's zero-energy edge modes (MZM's) emerging at the edge of the coupled chains. Correspondingly, the topological nature of the system is described by a phase diagram consisting of three different phases. The three phases are characterized by a topological winding number, $\mathcal{W}=1$, $2$ (with one and two MZM's: topological phases) and $\mathcal{W}=0$ (devoid of any MZM: trivial insulating phase).

arXiv:2103.00960•Mar 1, 2021•0 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
Using the density-matrix renormalization group, we determine the different topological phases and low-energy excitations of a time-reversal invariant topological superconducting (TRITOPS) wire with extended sss-wave superconductivity, Rashba spin-orbit coupling (SOC) and on-site repulsion UUU, under an externally applied Zeeman field JJJ. For the case in which JJJ is perpendicular to the SOC, the model describes a chain of Shiba impurities on top of a superconductor with extended superconductor pairing. We identify the different topological phases of the model at temperature T=0T=0T=0, and in particular we study the stability of the TRITOPS phase against the Zeeman field JJJ and the chemical potential \ensuremathμ\ensuremath{\mu}\ensuremathμ, for different values of UUU. In the case where the magnetic field JJJ is perpendicular to the SOC axis, the pair of Kramers degenerate Majorana zero modes at the edges of the system that exist for J=0J=0J=0, remain degenerate until a critical value of the magnetic field is reached. For JJJ parallel to the SOC and up to moderate values of UUU, the fractional spin projection \ensuremath⟨Sy\ensuremath⟩=1/4\ensuremath{\langle}{S}_{y}\ensuremath{\rangle}=1/4\ensuremath⟨Sy​\ensuremath⟩=1/4 at the ends, found for noninteracting wires at U=0U=0U=0, is recovered. In addition, the analytic expression that relates \ensuremath⟨Sy\ensuremath⟩\ensuremath{\langle}{S}_{y}\ensuremath{\rangle}\ensuremath⟨Sy​\ensuremath⟩ with JJJ for finite noninteracting chains is shown to be universal up to moderate values of UUU.
We consider a model proposed before for a time-reversal-invariant topological superconductor (TRITOPS) which contains a hopping term ttt, a chemical potential μ\muμ, an extended sss-wave pairing Δ\DeltaΔ and spin-orbit coupling λ\lambdaλ. We show that for ∣Δ∣=∣λ∣|\Delta|=|\lambda|∣Δ∣=∣λ∣, μ=t=0\mu=t=0μ=t=0, the model can be solved exactly defining new fermion operators involving nearest-neighbor sites. The many-body ground state is four-fold degenerate due to the existence of two zero-energy modes localized exactly at the first and the last site of the chain. These four states show entanglement in the sense that creating or annihilating a zero-energy mode at the first site is proportional to a similar operation at the last site. By continuity, this property should persist for general parameters. Using these results we correct some statements related with the so called time-reversal anomaly. Addition of a small hopping term for a chain with an even number of sites breaks the degeneracy and the ground state becomes unique with an even number of particles. We also consider a small magnetic field applied to one end of the chain. We compare the many-body excitation energies and spin projection along the spin-orbit direction for both ends of the chains with numerical results %for a small chain obtaining good agreement.
We study the ground state and low-energy subgap excitations of a finite wire of a time-reversal-invariant topological superconductor (TRITOPS) with spin-orbit coupling. We solve the problem analytically for a long chain of a specific one-dimensional lattice model in the electron-hole symmetric configuration and numerically for other cases of the same model. We present results for the spin density of excitations in long chains with an odd number of particles. The total spin projection along the axis of the spin-orbit coupling ${S}_{z}=\ifmmode\pm\else\textpm\fi{}1/2$ is distributed with fractions $\ifmmode\pm\else\textpm\fi{}1/4$ localized at both ends and shows even-odd alternation along the sites of the chain. We calculate the localization length of these excitations and find that it can be well approximated by a simple analytical expression. We show that the energy EEE of the lowest subgap excitations of the finite chain defines tunneling and entanglement between end states. We discuss the effect of a Zeeman coupling ${\mathrm{\ensuremath{\Delta}}}_{Z}$ on one of the ends of the chain only. For ${\mathrm{\ensuremath{\Delta}}}_{Z}<E$, the energy difference of excitations with opposite spin orientation is ${\mathrm{\ensuremath{\Delta}}}_{Z}/2$, consistent with a spin projection $\ifmmode\pm\else\textpm\fi{}1/4$. We argue that these physical features are not model dependent and can be experimentally observed in TRITOPS wires under appropriate conditions.
We consider a model proposed before for a time-reversal-invariant topological superconductor which contains a hopping term ttt, a chemical potential $\ensuremath{\mu}$, an extended sss-wave pairing $\mathrm{\ensuremath{\Delta}}$, and spin-orbit coupling $\ensuremath{\lambda}$. We show that for $|\mathrm{\ensuremath{\Delta}}|=|\ensuremath{\lambda}|$, $\ensuremath{\mu}=t=0$, the model has an exact analytical solution defining new fermion operators involving nearest-neighbor sites. The many-body ground state is fourfold degenerate due to the existence of two zero-energy modes localized exactly at the first and last sites of the chain. These four states show entanglement in the sense that creating or annihilating a zero-energy mode at the first site is proportional to a similar operation at the last site. By continuity, this property should persist for general parameters. Using these results, we discuss some statements related to the so-called time-reversal anomaly. The addition of a small hopping term ttt for a chain with an even number of sites breaks the degeneracy, and the ground state becomes unique with an even number of particles. We also consider a small magnetic field BBB applied to one end of the chain. We compare the many-body excitation energies and spin projection along the spin-orbit direction for both ends of the chains obtained treating ttt and BBB as small perturbations with numerical results in a short chain, obtaining good agreement.
Among the broad spectrum of systems predicted to exhibit topological superconductivity and Majorana fermions, one-dimensional wires with strong spin-orbit coupling provide one of the most promising experimental candidates. Here we investigate the fate of the topological superconducting phase in such wires when repulsive interactions are present. Using a combination of density matrix renormalization group, bosonization, and Hartree--Fock techniques, we demonstrate that while interactions degrade the bulk gap---consistent with recent results of Gangadharaiah et al.---they also greatly expand the parameter range over which the topological phase arises. In particular, we show that with interactions this phase can be accessed over a broader chemical potential window, thereby leading to greater immunity against disorder-induced chemical potential fluctuations in the wire. We also suggest that in certain wires strong interactions may allow Majorana fermions to be generated without requiring a magnetic field.
Evolution of Majorona zero-energy edge states in a T2=−1T^2 = -1T2=−1 symmetry protected 1D topological superconductor with dominant spin-orbit coupling
We consider a 1D topological superconductor (TSC) constructed by coupling a pair of Kitaev's Majorana chains with opposite spin configurations. Such a 1D lattice model is known to be protected by a T2=−1T^2 = -1T2=−1 time-reversal symmetry. Furthermore, we consider a modeled Rashba spin-orbit coupling on such a system of T2=−1T^2=-1T2=−1 time-reversal symmetric TSC. The Rashba spin-orbit coupling together with the chemical potential engineered the phase transitions of the edge states in the system and consequently the number of Majorona's zero-energy edge modes (MZM's) emerging at the edge of the coupled chains. Correspondingly, the topological nature of the system is described by a phase diagram consisting of three different phases. The three phases are characterized by a topological winding number, W=1\mathcal{W}=1W=1, 222 (with one and two MZM's: topological phases) and W=0\mathcal{W}=0W=0 (devoid of any MZM: trivial insulating phase).
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.