From one-dimensional charge conserving superconductors to the gapless Haldane phase

We develop a framework to analyze one-dimensional topological superconductors with charge conservation. In particular, we consider models with $N$ flavors of fermions and $(\mathbb{Z}_2)^N$ symmetry, associated with the conservation of the fermionic parity of each flavor. For a single flavor, we recover the result that a distinct topological phase with exponentially localized zero modes does not exist due to absence of a gap to single particles in the bulk. For $N>1$, however, we show that the ends of the system can host low-energy, exponentially-localized modes. The analysis can readily be generalized to systems in other symmetry classes. To illustrate these ideas, we focus on lattice models with $SO\left(N\right)$ symmetric interactions, and study the phase transition between the trivial and the topological gapless phases using bosonization and a weak-coupling renormalization group analysis. As a concrete example, we study in detail the case of $N=3$. We show that in this case, the topologically non-trivial superconducting phase corresponds to a gapless analogue of the Haldane phase in spin-1 chains. In this phase, although the bulk is gapless to single particle excitations, the ends host spin-$1/2$ degrees of freedom which are exponentially localized and protected by the spin gap in the bulk. We obtain the full phase diagram of the model numerically, using density matrix renormalization group calculations. Within this model, we identify the self-dual line studied by Andrei and Destri [Nucl. Phys. B, 231(3), 445-480 (1984)], as a first-order transition line between the gapless Haldane phase and a trivial gapless phase. This allows us to identify the propagating spin-$1/2$ kinks in the Andrei-Destri model as the topological end-modes present at the domain walls between the two phases

Matrix Product Operators, Matrix Product States, and ab initio Density Matrix Renormalization Group algorithms

Current descriptions of the ab initio DMRG algorithm use two superficially different languages: an older language of the renormalization group and renormalized operators, and a more recent language of matrix product states and matrix product operators. The same algorithm can appear dramatically different when written in the two different vocabularies. In this work, we carefully describe the translation between the two languages in several contexts. First, we describe how to efficiently implement the ab-initio DMRG sweep using a matrix product operator based code, and the equivalence to the original renormalized operator implementation. Next we describe how to implement the general matrix product operator/matrix product state algebra within a pure renormalized operator-based DMRG code. Finally, we discuss two improvements of the ab initio DMRG sweep algorithm motivated by matrix product operator language: Hamiltonian compression, and a sum over operators representation that allows for perfect computational parallelism. The connections and correspondences described here serve to link the future developments with the past, and are important in the efficient implementation of continuing advances in ab initio DMRG and related algorithms.Comment: 35 pages, 10 figure

Criticality and entanglement in non-unitary quantum circuits and tensor networks of non-interacting fermions

Models for non-unitary quantum dynamics, such as quantum circuits that include projective measurements, have been shown to exhibit rich quantum critical behavior. There are many complementary perspectives on this behavior. For example, there is a known correspondence between d-dimensional local non-unitary quantum circuits and tensor networks on a D=(d+1)-dimensional lattice. Here, we show that in the case of systems of non-interacting fermions, there is furthermore a full correspondence between non-unitary circuits in d spatial dimensions and unitary non-interacting fermion problems with static Hermitian Hamiltonians in D=(d+1) spatial dimensions. This provides a powerful new perspective for understanding entanglement phases and critical behavior exhibited by non-interacting circuits. Classifying the symmetries of the corresponding non-interacting Hamiltonian, we show that a large class of random circuits, including the most generic circuits with randomness in space and time, are in correspondence with Hamiltonians with static spatial disorder in the ten Altland-Zirnbauer symmetry classes. We find the criticality that is known to occur in all of these classes to be the origin of the critical entanglement properties of the corresponding random non-unitary circuit. To exemplify this, we numerically study the quantum states at the boundary of Haar-random Gaussian fermionic tensor networks of dimension D=2 and D=3. We show that the most general such tensor network ensemble corresponds to a unitary problem of non-interacting fermions with static disorder in Altland-Zirnbauer symmetry class DIII, which for both D=2 and D=3 is known to exhibit a stable critical metallic phase. Tensor networks and corresponding random non-unitary circuits in the other nine Altland-Zirnbauer symmetry classes can be obtained from the DIII case by implementing Clifford algebra extensions for classifying spaces.Comment: (25+14) pages, 19 figure

High order magnon bound states in the quasi-one-dimensional antiferromagnet α\alpha-NaMnO2_2

Here we report on the formation of two and three magnon bound states in the quasi-one-dimensional antiferromagnet $\alpha$-NaMnO$_2$, where the single-ion, uniaxial anisotropy inherent to the Mn$^{3+}$ ions in this material provides a binding mechanism capable of stabilizing higher order magnon bound states. While such states have long remained elusive in studies of antiferromagnetic chains, neutron scattering data presented here demonstrate that higher order $n>2$ composite magnons exist, and, specifically, that a weak three-magnon bound state is detected below the antiferromagnetic ordering transition of NaMnO$_2$. We corroborate our findings with exact numerical simulations of a one-dimensional Heisenberg chain with easy-axis anisotropy using matrix-product state techniques, finding a good quantitative agreement with the experiment. These results establish $\alpha$-NaMnO$_2$ as a unique platform for exploring the dynamics of composite magnon states inherent to a classical antiferromagnetic spin chain with Ising-like single ion anisotropy.Comment: 5 pages, 4 figure

Evidence of topological superconductivity in planar Josephson junctions

Majorana zero modes are quasiparticle states localized at the boundaries of topological superconductors that are expected to be ideal building blocks for fault-tolerant quantum computing. Several observations of zero-bias conductance peaks measured in tunneling spectroscopy above a critical magnetic field have been reported as experimental indications of Majorana zero modes in superconductor/semiconductor nanowires. On the other hand, two dimensional systems offer the alternative approach to confine Ma jorana channels within planar Josephson junctions, in which the phase difference {\phi} between the superconducting leads represents an additional tuning knob predicted to drive the system into the topological phase at lower magnetic fields. Here, we report the observation of phase-dependent zero-bias conductance peaks measured by tunneling spectroscopy at the end of Josephson junctions realized on a InAs/Al heterostructure. Biasing the junction to {\phi} ~ {\pi} significantly reduces the critical field at which the zero-bias peak appears, with respect to {\phi} = 0. The phase and magnetic field dependence of the zero-energy states is consistent with a model of Majorana zero modes in finite-size Josephson junctions. Besides providing experimental evidence of phase-tuned topological superconductivity, our devices are compatible with superconducting quantum electrodynamics architectures and scalable to complex geometries needed for topological quantum computing.Comment: main text and extended dat