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arXiv:1806.06104 · Physical review. B./Physical review. B

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

Armando A. Aligia, Liliana Arrachea

Published Nov 20, 2018• 22 citations
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Abstract

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.

DOI
10.1103/physrevb.98.174507

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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 Sz=\ifmmode±\else\textpm\fi1/2{S}_{z}=\ifmmode\pm\else\textpm\fi{}1/2Sz​=\ifmmode±\else\textpm\fi1/2 is distributed with fractions \ifmmode±\else\textpm\fi1/4\ifmmode\pm\else\textpm\fi{}1/4\ifmmode±\else\textpm\fi1/4 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 \ensuremathΔZ{\mathrm{\ensuremath{\Delta}}}_{Z}\ensuremathΔZ​ on one of the ends of the chain only. For \ensuremathΔZ<E{\mathrm{\ensuremath{\Delta}}}_{Z}<E\ensuremathΔZ​<E, the energy difference of excitations with opposite spin orientation is \ensuremathΔZ/2{\mathrm{\ensuremath{\Delta}}}_{Z}/2\ensuremathΔZ​/2, consistent with a spin projection \ifmmode±\else\textpm\fi1/4\ifmmode\pm\else\textpm\fi{}1/4\ifmmode±\else\textpm\fi1/4. 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 (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.
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 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.
We consider a one-dimensional, time-reversal-invariant system with attractive interactions and spin-orbit coupling. Such a system is gapless due to the strong quantum fluctuations of the superconducting order parameter. However, we show that a sharply defined topological phase with protected, exponentially localized edge states exists. If one of the spin components is conserved, the protection of the edge modes can be understood as a consequence of the presence of a spin gap. In the more general case, the localization of the edge states arises from a gap to single-particle excitations in the bulk. We consider specific microscopic models and demonstrate both analytically and numerically (using density matrix renormalization group calculations) that they can support the topologically nontrivial phase.
In this work we study a two-dimensional topological superconductor on a square lattice. It contains hopping, spin-orbit coupling and a time reversal symmetry breaking Zeeman term. This realizes, in combination with a singlet pairing term, an effective triplet pairing that leads to a topological superconductor. As previously found numerically, the transitions are seen in the entanglement entropy as cusps as a function of model parameters. We complement this through studying the entanglement entropy analytically by keeping only its most important components. Our study is based on the intuition that the number of Fermi surfaces in the system controls the topological invariant. With our approximate expression for the entanglement entropy we are able to extract the divergent entanglement entropy derivative close to the phase transition.
The manifestations of topological effects in finite antiferromagnetic Heisenberg chains are examined by density-matrix renormalization-group techniques. We find that the difference between integer and half-integer spin chains shows up in ground-state energy per site when the length of the spin chain is longer than \ensuremath{\sim}\ensuremath{\xi}, where $\ensuremath{\xi}\ensuremath{\sim}\mathrm{exp}(\ensuremath{\pi}S)$ is a spin-spin correlation length, for spin magnitudes S up to 5/2.5/2.5/2. For open chains with spin magnitudes S=5/2S=5/2S=5/2 to S=5,S=5,S=5, we verify that end states with fractional spin quantum numbers ${S}^{\ensuremath{'}}$ exist and are visible in the structure of the low-energy excitation spectrum even when the chain length is much smaller than the correlation length \ensuremath{\xi}. The robustness of end states points to the possibility of the existence of general hidden topological order in quantum spin chains with spin magnitudes S> 1.S>~1.S> 1.