We investigate one-dimensional charge conserving, spin-singlet (SSS) and spin-triplet (STS) superconductors in the presence of boundary fields. In systems with Open Boundary Conditions (OBC) it has been demonstrated that STS display a four-fold topol
ogical degeneracy, protected by the $mathbb{Z}_2$ symmetry which reverses the spins of all fermions, whereas SSS are topologically trivial. In this work we show that it is not only the type of the bulk superconducting instability that determines the eventual topological nature of a phase, but rather the interplay between bulk and boundary properties. In particular we show by means of the Bethe Ansatz technique that SSS may as well be in a $mathbb{Z}_2$-protected topological phase provided suitable twisted open boundary conditions ${widehat{OBC}}$ are imposed. More generally, we find that depending on the boundary fields, a given superconductor, either SSS or STS, may exhibits several types of phases such as topological, mid-gap and trivial phases; each phase being characterized by a boundary fixed point which which we determine. Of particular interest are the mid-gap phases which are stabilized close to the topological fixed point. They include both fractionalized phases where spin-$frac{1}{4}$ bound-states are localized at the two edges of the system and un-fractionalized phases where a spin-$frac{1}{2}$ bound-state is localized at either the left or the right edge.
We extend the general formalism discussed in the previous paper [A. B. Culver and N. Andrei, Phys. Rev. B 103, 195106 (2021)] to two models with charge fluctuations: the interacting resonant level model and the Anderson impurity model. In the interac
ting resonant level model, we find the exact time-evolving wavefunction and calculate the steady state impurity occupancy to leading order in the interaction. In the Anderson impurity model, we find the nonequilibrium steady state for small or large Coulomb repulsion $U$, and we find that the steady state current to leading order in $U$ agrees with a Keldysh perturbation theory calculation.
Charge conserving spin singlet and spin triplet superconductors in one dimension are described by the $U(1)$ symmetric Thirring Hamiltonian. We solve the model with open boundary conditions on the a finite line segment by means of the Bethe Ansatz. W
e show that the ground state displays a fourfold degeneracy when the bulk is in the spin triplet superconducting phase. This degeneracy corresponds to the existence of zero energy boundary bound states localized at the edges which may be interpreted, in the light of the previous semi-classical analysis due to Kesselman and Berg cite{Keselman2015}, as resulting from the existence of fractional spin $pm 1/4$ localized at the two edges of the system.
We present a method for calculating the time-dependent many-body wavefunction that follows a local quench. We apply the method to the voltage-driven nonequilibrium Kondo model to find the exact time-evolving wavefunction following a quench where the
dot is suddenly attached to the leads at $t=0$. The method, which does not use Bethe ansatz, also works in other quantum impurity models (we include results for the interacting resonant level and the Anderson impurity model) and may be of wider applicability. In the particular case of the Kondo model, we show that the long-time limit (with the system size taken to infinity first) of the time-evolving wavefunction is a current-carrying nonequilibrium steady state that satisfies the Lippmann-Schwinger equation. We show that the electric current in the time-evolving wavefunction is given by a series expression that can be expanded either in weak coupling or in strong coupling, converging to all orders in the steady-state limit in either case. The series agrees to leading order with known results in the well-studied regime of weak antiferromagnetic coupling and also reveals another universal regime of strong ferromagnetic coupling, with Kondo temperature $T_K^{(F)} = D e^{-frac{3pi^2}{8} rho |J|}$ ($J<0$, $rho|J|toinfty$). In this regime, the differential conductance $dI/dV$ reaches the unitarity limit $2e^2/h$ asymptotically at large voltage or temperature.
In the previous paper, we found a series expression for the average electric current following a quench in the nonequilibrium Kondo model driven by a bias voltage. Here, we evaluate the steady state current in the regimes of strong and weak coupling.
We obtain the standard leading order results in the usual weak antiferromagnetic regime, and we also find a new universal regime of strong ferromagnetic coupling with Kondo temperature $T_K = D e^{frac{3pi^2}{8} rho J}$. In this regime, the differential conductance $dI/dV$ reaches the unitarity limit $2e^2/h$ asymptotically at large voltage or temperature.
We present here the details of a method [A. B. Culver and N. Andrei, Phys. Rev. B 103, L201103 (2021)] for calculating the time-dependent many-body wavefunction that follows a local quench. We apply the method to the voltage-driven nonequilibrium Kon
do model to find the exact time-evolving wavefunction following a quench where the dot is suddenly attached to the leads at $t=0$. The method, which does not use Bethe ansatz, also works in other quantum impurity models and may be of wider applicability. We show that the long-time limit (with the system size taken to infinity first) of the time-evolving wavefunction of the Kondo model is a current-carrying nonequilibrium steady state that satisfies the Lippmann-Schwinger equation. We show that the electric current in the time-evolving wavefunction is given by a series expression that can be expanded either in weak coupling or in strong coupling, converging to all orders in the steady-state limit in either case. The series agrees to leading order with known results in the well-studied regime of weak antiferromagnetic coupling and also reveals a universal regime of strong ferromagnetic coupling with Kondo temperature $T_K^{(F)} = D e^{-frac{3pi^2}{8} rho |J|}$ ($J<0$, $rho|J|toinfty$). In this regime, the differential conductance $dI/dV$ reaches the unitarity limit $2e^2/h$ asymptotically at large voltage or temperature.
Quantum impurity models are prevalent throughout many body physics, providing some prime examples of strongly correlated systems. Aside from being of great interest in themselves they can provide deep insight into the effects of strong correlations i
n general. The classic example is the Kondo model wherein a magnetic impurity is screened at low energies by a non interacting metallic bath. Here we consider a magnetic impurity coupled to a quantum wire with pairing interaction which dynamically generates a mass gap. Using Bethe Ansatz we solve the system exactly finding that it exhibits both screened and unscreened phases for an antiferromagnetic impurity. We determine the ground state density of states and magnetization in both phases as well as the excitations. In contrast to the well studied case of magnetic impurities in superconductors we find that there are no intragap bound states in the spectrum. The phase transition is not associated to a level crossing but with quantum fluctuations.
Driven by breakthroughs in experimental and theoretical techniques, the study of non-equilibrium quantum physics is a rapidly expanding field with many exciting new developments. Amongst the manifold ways the topic can be investigated, one dimensiona
l system provide a particularly fine platform. The trifecta of strongly correlated physics, powerful theoretical techniques and experimental viability have resulted in a flurry of research activity over the last decade or so. In this review we explore the non equilibrium aspects of one dimensional systems which are integrable. Through a number of illustrative examples we discuss non equilibrium phenomena which arise in such models, the role played by integrability and the consequences these have for more generic systems.
A classic example of a quantum quench concerns the release of a interacting Bose gas from an optical lattice. The local properties of quenches such as this have been extensively studied however the global properties of these non-equilibrium quantum s
ystems have received far less attention. Here we study several aspects of global non-equilibrium behavior by calculating the amount of work done by the quench as measured through the work distribution function. Using Bethe Ansatz techniques we determine the Loschmidt amplitude and work distribution function of the Lieb-Liniger gas after it is released from an optical lattice. We find the average work and its universal edge exponents from which we determine the long time decay of the Loshcmidt echo and highlight striking differences caused by the the interactions as well as changes in the geometry of the system. We extend our calculation to the attractive regime of the model and show that the system exhibits properties similar to the super Tonks-Girardaeu gas. Finally we examine the prominent role played by bound states in the work distribution and show that, with low probability, they allow for work to be extracted from the quench.
Proceedings of the workshop Boundary and Defect Conformal Field Theory: Open Problems and Applications, Chicheley Hall, Buckinghamshire, UK, 7-8 Sept. 2017.
