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

Armando A. Aligia, Alberto Camjayi

Published May 10, 2019• 0 citations
OpenAlex

Abstract

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.

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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.

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

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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 $U$.

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

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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.

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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
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 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.
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 $\ensuremath{\mu}$, 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 $\ensuremath{\langle}{S}_{y}\ensuremath{\rangle}=1/4$ at the ends, found for noninteracting wires at U=0U=0U=0, is recovered. In addition, the analytic expression that relates $\ensuremath{\langle}{S}_{y}\ensuremath{\rangle}$ with JJJ for finite noninteracting chains is shown to be universal up to moderate values of UUU.
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.
A time-reversal invariant Kitaev-type model is introduced in which spins (Dirac matrices) on the square lattice interact via anisotropic nearest-neighbor and next-nearest-neighbor exchange interactions. The model is exactly solved by mapping it onto a tight-binding model of free Majorana fermions coupled with static Z2{\mathbb{Z}}_{2}Z2​ gauge fields. The Majorana fermion model can be viewed as a model of time-reversal-invariant superconductor and is classified as a member of symmetry class DIII in the Altland-Zirnbauer classification. The ground-state phase diagram has two topologically distinct gapped phases which are distinguished by a Z2{\mathbb{Z}}_{2}Z2​ topological invariant. The topologically nontrivial phase supports both a Kramers' pair of gapless Majorana edge modes at the boundary and a Kramers' pair of zero-energy Majorana states bound to a 0-flux vortex in the $\ensuremath{\pi}$-flux background. Power-law decaying correlation functions of spins along the edge are obtained by taking the gapless Majorana edge modes into account. The model is also defined on the one-dimension ladder, in which case again the ground-state phase diagram has Z2{\mathbb{Z}}_{2}Z2​ trivial and nontrivial phases.
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.
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).