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arXiv:1802.02316 · Physical review. B./Physical review. B

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

Anna Keselman, Erez Berg, P. Azaria

Published Dec 3, 2018• 34 citations
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Abstract

We develop a framework to analyze the topological properties of one-dimensional systems with charge conservation and tendency towards topological superconducting order. 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 $N=1$, and with no other symmetry other than charge conservation, we recover the result that there is no distinct topological phase with exponentially localized zero modes. For $N>1$, however, we show that the ends of the system can host low-energy, exponentially-localized modes. 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 $N=3$. In this case, the topologically nontrivial superconducting phase corresponds to a gapless analog of the Haldane phase in spin-1 chains. In this phase, although the bulk hosts gapless modes, corresponding to composite fermionic excitations with an enlarged Fermi surface, 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 using density matrix renormalization group calculations. Within this model, we identify the self-dual line studied by Andrei and Destri [Nucl. Phys. B 231, 445 (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 at the domain walls between the two phases.

DOI
10.1103/physrevb.98.214501

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We develop a framework to analyze the topological properties of one-dimensional systems with charge conservation and tendency towards topological superconducting order. In particular, we consider models with NNN flavors of fermions and (Z2)N{({\mathbb{Z}}_{2})}^{N}(Z2​)N symmetry, associated with the conservation of the fermionic parity of each flavor. For N=1N=1N=1, and with no other symmetry other than charge conservation, we recover the result that there is no distinct topological phase with exponentially localized zero modes. For N>1N>1N>1, however, we show that the ends of the system can host low-energy, exponentially-localized modes. To illustrate these ideas, we focus on lattice models with SO(N)SO\left(N\right)SO(N) 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=3N=3N=3. In this case, the topologically nontrivial superconducting phase corresponds to a gapless analog of the Haldane phase in spin-1 chains. In this phase, although the bulk hosts gapless modes, corresponding to composite fermionic excitations with an enlarged Fermi surface, the ends host spin-1/21/21/2 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 using density matrix renormalization group calculations. Within this model, we identify the self-dual line studied by Andrei and Destri [Nucl. Phys. B 231, 445 (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-1/21/21/2 kinks in the Andrei-Destri model as the topological end modes at the domain walls between the two phases.
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.
Understanding the interplay between symmetries and phases lies at the heart of characterizing states of matter. The conventional Landau's paradigm associates local ordered states with symmetry breaking, but this paradigm fails to include gapped topological phases, which can only be distinguished by the global topological invariant and the present of gapless edge modes at open boundaries. In this thesis, variety of states of matter is enriched from two aspects: a new paradigm of phases that incorporates both symmetry broken phase and topological phase, and exotic topological phases that are realized in different degree of freedoms of the same system. The two aspects are explored by studying corresponding one dimensional fermionic models. Besides gapped phases, exotic quasiparticle interferences of gapless topological systems are also investigated. \n \nThe first chapter serves as a theoretical basis for the next two chapters by giving an introduction to topological phases in one dimension. In particular, a bosonization method is introduced to describe topological degeneracy and edge zero modes, which is favoured by the ability to deal with interactions as well as to uncover the phenomena of spin-charge separation. \n \nIn the second chapter a new paradigm of phases is proposed and investigated that incorporates both symmetry broken phases and symmetry-protected topological phases. Through exact-solvable models bond spin density wave(BSDW) states are proposed as candidates for this new paradigm, which are ZZ2\\ZZ_2ZZ2​ symmetry breaking due to spontaneous dimerization and yet each is topological nontrivial with edge zero modes of fractional fermions. More general lattice models are constructed, and it is shown that BSDW exists in a considerable regime of parameters for a family of lattice models, from both mean-field analysis and bosonization. \n \nIn the third chapter, we give an effective field description for spin-charge separated edge modes consisting of both Majorana modes and fractional fermions, and demonstrate how to harvest both simultaneously in a spinless fermionic ladder with particle-hole symmetry and single-chain fermion parity. Further, the idea of spin-charge separated edge modes is applied to reinterpret the prominent Haldane phase of spin-1 chain in its equivalent spin-frac12\\frac{1}{2}frac12 fermionic model. \n \nIn contrast to the above two theoretically oriented topics that aim to add variety to topological phases, the third topic establishes a bridge between abstract concepts for topological systems to remarkable features that can be detected in experiments. The last chapter reveals how topological charges for topological Fermi points, through exhibiting exotic quasiparticle interference in the presence of impurities, could be visualized in STS measurements. \nThe generalized joint density of states(GJDOS) is applied to address effects from the geometry of band dispersion and topological charges separately. Both analytical results and numerical simulation suggest distinct and robust quasiparticle interference features as indicators for topological charges that can be accessed by FT-STS experiments.
The low energy behavior of a huge variety of one-dimensional interacting spinful fermionic systems exhibits spin-charge separation, described in the continuum limit by two sine-Gordon models decoupled in the charge and spin channels. Interaction is known to induce, besides the gapless Luttinger liquid phase, eight possible gapped phases, among which are the Mott, Haldane, charge-/spin-density, and bond-ordered wave insulators, and the Luther Emery liquid. Here we prove that some of these physically distinct phases have nontrivial topological properties, notably the presence of degenerate protected edge modes with fractionalized charge/spin. Moreover, we show that the eight gapped phases are in one-to-one correspondence with the symmetry-protected topological (SPT) phases classified by group cohomology theory in the presence of particle-hole symmetry P. The latter result is also exploited to characterize SPT phases by measurable nonlocal order parameters which follow the system evolution to the quantum phase transition. The implications on the appearance of exotic orders in the class of microscopic Hubbard Hamiltonians, possibly without P symmetry at higher energies, are discussed.
We reexamine the one-dimensional spin-1 XXZXXZXXZ model with on-site uniaxial single-ion anisotropy as to the appearance and characterization of the symmetry-protected topological Haldane phase. By means of large-scale density-matrix renormalization group (DMRG) calculations the central charge can be determined numerically via the von Neumann entropy, from which the ground-state phase diagram of the model can be derived with high precision. The nontrivial gapped Haldane phase shows up in between the trivial gapped even Haldane and N\'eel phases, appearing at large single-ion and spin-exchange interaction anisotropies, respectively. We furthermore carve out a characteristic degeneracy of the lowest entanglement level in the topological Haldane phase, which is determined using a conventional finite-system DMRG technique with both periodic and open boundary conditions. Defining the spin and neutral gaps in analogy to the single-particle and neutral gaps in the intimately connected extended Bose-Hubbard model, we show that the excitation gaps in the spin model qualitatively behave just as for the bosonic system. We finally compute the dynamical spin structure factor in the three different gapped phases and find significant differences in the intensity maximum which might be used to distinguish these phases experimentally.
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 fourfold topological degeneracy, protected by the Z2{\mathbb{Z}}_{2}Z2​ symmetry, which reverses the spins of all fermions, whereas SSS are topologically trivial. In this paper 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 Z2{\mathbb{Z}}_{2}Z2​-protected topological phase provided suitable ``twisted'' open boundary conditions $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{OBC}}$ are imposed. More generally, we find that depending on the boundary fields, a given superconductor, either SSS or STS, may exhibit several types of phases such as topological, mid-gap, and trivial phases; each phase being characterized by a boundary fixed point, which we determine. Of particular interest are the mid-gap phases, which are stabilized close to the topological fixed point. They include fractionalized phases where spin-1/41/41/4 bound states are localized at the two edges of the system and unfractionalized phases where a spin-1/21/21/2 bound state is localized at either the left or the right edge.
We investigate the ground-state properties of a recently proposed model for a topological Kondo insulator in one dimension (i.e., the ppp-wave Kondo-Heisenberg lattice model) by means of the density-matrix renormalization-group method. The nonstandard Kondo interaction in this model is different from the usual (i.e., local) Kondo interaction in that the localized spins couple to the ``ppp-wave'' spin density of conduction electrons, inducing a topologically nontrivial insulating ground state. Based on the analysis of the charge- and spin-excitation gaps, the string order parameter, and the spin profile in the ground state, we show that, at half filling and low energies, the system is in the Haldane phase and hosts topologically protected spin-1/2 end states. Beyond its intrinsic interest as a useful ``toy model'' to understand the effects of strong correlations on topological insulators, we show that the ppp-wave Kondo-Heisenberg model could be experimentally implemented in $p\text{\ensuremath{-}}\mathrm{band}$ optical lattices loaded with ultracold Fermi gases.