We present a detailed study of the fidelity, the entanglement entropy, and the entanglement spectrum, for a dimerized chain of spinless fermions---a simplified Su-Schrieffer-Heeger (SSH) model---with open boundary conditions which is a well-known example for a model supporting a symmetry protected topological (SPT) phase. In the non-interacting case the Hamiltonian matrix is tridiagonal and the eigenvalues and -vectors can be given explicitly as a function of a single parameter which is known analytically for odd chain lengths and can be determined numerically in the even length case. From a scaling analysis of these data for essentially semi-infinite chains we obtain the fidelity susceptibility and show that it contains a boundary contribution which is different in the topologically ordered than in the topologically trivial phase. For the entanglement spectrum and entropy we confirm predictions from massive field theory for a block in the middle of an infinite chain but also consider blocks containing the edge of the chain. For the latter case we show that in the SPT phase additional entanglement---as compared to the trivial phase---is present which is localized at the boundary. Finally, we extend our study to the dimerized chain with a nearest-neighbour interaction using exact diagonalization, Arnoldi, and density-matrix renormalization group methods and show that a phase transition into a topologically trivial charge-density wave phase occurs.
Exploiting the possibility of temporal variation of the winding number, we have prepared a SSH chain in its {\it stroboscopic} topological state, starting from the trivial one, by application of a periodic perturbation. The periodic driving, we employ here, is adiabatically switched on to break the particle-hole symmetry and generate a chiral mass term in the effective Floquet Hamiltonian; consequently the Floquet Hamiltonian also gets deformed without crossing the gapless quantum critical point. The particle hole symmetry is subsequently restored in the Floquet Hamiltonian by adiabatically sw…
Universal scaling of spatially extended zero modes in inhomogeneous SSH chains
Laboratoire d’Annecy de Physique Théorique (LAPTh), CNRS, Université Savoie Mont-Blanc（萨瓦蒙布朗大学）；Institut Denis-Poisson, CNRS, Université de Tours–Université d’Orléans（图尔大学与奥尔良大学）；Centre de Recherches Mathématiques, Université de Montréal（蒙特利尔大学）；Département de Physique, Université de Montréal（蒙特利尔大学）
We construct a two-leg ladder dimer model by using two orbitals instead of one in Su-Schrieffer-Heeger (SSH) dimer chain and find out a chiral symmetry in it. In this model, the otherwise-hidden additional chiral symmetry allows us to define two chiral massless fermions and show that there exist interesting topological states characterized by $\mathbb{Z}\oplus\mathbb{Z}$ and corresponding zero mode edge states. Our complete phase diagram reveals that there is an anomalous topologically nontrivial region in addition to the normal one similar to that of SSH model. The anomalous region features t…
Symmetry-protected topological (SPT) phases are short-range entangled phases of matter with a non-local order parameter which are preserved under a local symmetry group. Here, by using unsupervised learning algorithm, namely the diffusion maps, we demonstrate that can differentiate between symmetry broken phases and topologically ordered phases, and between non-trivial topological phases in different classes. In particular, we show that the phase transitions associated with these phases can be detected in different bosonic and fermionic models in one dimension. This includes the interacting SS…
We use an entanglement measure that respects the superselection of particle number to study the non-local properties of symmetry-protected topological edge states. Considering half-filled M-leg Su-Schrieffer-Heeger (SSH) ladders as an example, we show that the topological properties and the operational entanglement extractable from the boundaries are intimately connected. Topological phases with at least two filled edge states have the potential to realize genuine, non-bipartite, many-body entanglement which can be transferred to a quantum register. The entanglement is extractable when the fil…
We investigate topological signatures in the short-time non-equilibrium dynamics of symmetry protected topological (SPT) systems starting from initial states which break the protecting symmetry. Naïvely, one might expect that topology loses meaning when a protecting symmetry is broken. Defying this intuition, we illustrate, in an interacting Su-Schrieffer-Heeger (SSH) model, how this combination of symmetry breaking and quench dynamics can give rise to both single-particle and many-body signatures of topology. From the dynamics of the symmetry broken state, we find that we are able to dynamica…
The search for topological insulators has been actively promoted in the field of condensed matter physics for further development in energy-efficient information transmission and processing. In this context, recent studies have revealed that not only electrons but also bosonic particles such as magnons can construct edge states carrying nontrivial topological invariants. Here we demonstrate topological triplon bands in the spin-1/2 two-dimensional dimerized quantum antiferromagnet Ba$_2$CuSi$_2$O$_6$Cl$_2$, which is closely related to a pseudo-one-dimensional variant of the Su-Schrieffer-Heege…
We investigate entanglement naturally present in the 1D topologically ordered phase protected with the on-site symmetry group of an octahedron as a potential resource for teleportation-based quantum computation. We show that, as long as certain characteristic lengths are finite, all its ground states have the capability to implement any unit-fidelity one-qubit gate operation asymptotically as a key computational building block. This feature is intrinsic to the entire phase, in that perfect gate fidelity coincides with perfect string order parameters under a state-insensitive renormalization pr…
Dynamical phase transitions (DPT) are characterized by nonanalytical time evolution of the dynamical free energy. For general 2-band systems in one and two dimensions (eg. SSH model, Kitaev-chain, Haldane model, p+ip superconductor, etc.), we show that the time evolution of the dynamical free energy is crucially affected by the ground state topology of both the initial and final Hamiltonians, implying DPTs when the topology is changed under the quench. Similarly to edge states in topological insulators, DPTs can be classified as being topologically protected or not. In 1D systems the number of…
The Su-Schrieffer-Heeger (SSH) model, containing dimerized hopping and a constant onsite energy, has become a paradigmatic model for one-dimensional topological phases, soliton excitations and fractionalized charge in the presence of chiral symmetry. Motivated by the recent developments in engineering artificial lattices, we study an alternative model where hopping is constant but the onsite energy is dimerized. We find that it has a non-symmorphic chiral symmetry and supports topologically distinct phases described by a $\mathbb{Z}_{2}$ invariant $ν$. In the case of multimode ribbon we also f…
In the long-range Su-Schriffer-Heeger (SSH) model, in which the next nearest-neighbor hopping is considered, there exhibits a rich topological phase diagram that contains winding numbers $w=0, 1$, and $2$. In the presence of disorder, the change in mean winding number with various disorder strengths is numerically calculated. We find that the disorder drives phase transitions: $w=0 \rightarrow 1, 0 \rightarrow 2 $, $1 \rightarrow 2$ and $2 \rightarrow 1$, in which the non-zero winding numbers driven by disorder are called the topological Anderson insulating (TAI) phases. The transition mechani…
The Su-Schrieffer-Heeger (SSH) model describes a one-dimensional $Z_{2}$ topological insulator, which has two topological distinct phases corresponding to two different dimerizations. When spin-orbit coupling is introduced into the SSH model, we find the structure of the Bloch bands can be greatly changed, and most interestingly, a new topological phase with single zero-energy bound state which exhibits non-Abelian statistics at each end emerges, which suggests that a new topological invariant is needed to fully classify all phases. In a comparatively large range of parameters, we find that sp…
The fidelity susceptibility is a tool for studying quantum phase transitions in the Hermitian condensed matter systems. Recently, it has been generalized with the biorthogonal basis for the non-Hermitian quantum systems. From the general perturbation description with the constraint of parity-time (PT) symmetry, we show that the fidelity $\mathcal{F}$ is always real for the PT-unbroken states. For the PT-broken states, the real part of the fidelity susceptibility $\mathrm{Re}[\mathcal{X}_F]$ is corresponding to considering both the PT partner states, and the negative infinity is explored by the…
In this letter, we propose an application of String Order Parameter (SOP), commonly used in quantum spin systems, to identify symmetry-protected topological phase (SPT) in fermionic systems in the example of the dimerized fermionic chain. As a generalized form of dimerized model, we consider a one-dimensional spin-1/2 XX model with alternating spin couplings. We employ Jordan-Wigner fermionization to map this model to the spinless Su-Schrieffer-Heeger fermionic model (SSH) with generalized hopping signs. We demonstrate a phase transition between a trivial insulating phase and the Haldane phase…
Topologically Protected Learning from Exceptional Point Braiding: Toward Braid Programming
University of Bamenda（班达马大学）；Higher Teachers Training College, University of Bamenda（班达马大学高等教师培训学院）；Department of Physics, Faculty of Science, University of Dschang（杜尚格大学理学院物理系）；Unité de Recherche de Matière Condensée, d’Électronique et de Traitement de Signal (URMACETS)（凝聚态物质、电子与信号处理研究组）；Promotion Centre of Research for Technological Advancement and Sustainable Development (PCR-TASD)（促进技术进步与可持续发展研究中心）；Quantum Materials and Computing Group (QMaCG)（量子材料与计算小组）；Université des Sciences et Techniques de Masuku（马苏库科学技术大学）；Laboratoire d’Optique, Laser et Applications（光学、激光与应用实验室）
We study the phase diagram of the anisotropic spin-1 Heisenberg chain with single ion anisotropy (D) using a ground-state fidelity approach. The ground-state fidelity and its corresponding susceptibility are calculated within the quantum renormalization group scheme where we obtained the renormalization of fidelity preventing to calculate the ground state. Using this approach, the phase boundaries between the antiferromagnetic Néel, Haldane and large-D phases are obtained for the whole phase diagram, which justifies the application of quantum renormalization group to trace the symmetery protec…
Non-Abelian Statistics for Bosonic Symmetry-Protected Topological Phases
International Center for Quantum Materials and School of Physics, Peking University（北京大学量子材料科学中心及物理学院）；International Quantum Academy, Shenzhen（深圳国际量子研究院）；Hefei National Laboratory（合肥国家实验室）
Nontrivial topology in physical systems is the driving force behind many phenomena. Notably, phases of matter must be classified in part by their topological properties. Phases with topological order (TO), such as the fractional quantum Hall effect (QHE), can support anyonic excitations obeying fractional statistics with potential application in topological quantum computing. States lacking intrinsic TO can still be in topological phases provided certain symmetries are imposed. On their own, these symmetry-protected phases do not support anyons, but they can still have other interesting featur…
We show that mutual statistics between quantum particles can be tuned to generate emergent novel few particle quantum mechanics for the boundary modes of symmetry-protected topological phases of matter. As a concrete setting, we study a system of pseudofermions, defined as quantum particles with tunable algebra, which lie on two distinct Su-Schrieffer-Heeger (SSH) chains. We find that as the mutual statistics of the particles are tuned -- the boundary modes present in the two chains gets non-trivially entangled showing a sudden jump in their mutual entanglement entropy. We further show that, s…
We investigate the Loschmidt amplitude and dynamical quantum phase transitions in multiband one dimensional topological insulators. For this purpose we introduce a new solvable multiband model based on the Su-Schrieffer-Heeger model, generalized to unit cells containing many atoms but with the same symmetry properties. Such models have a richer structure of dynamical quantum phase transitions than the simple two-band topological insulator models typically considered previously, with both quasiperiodic and aperiodic dynamical quantum phase transitions present. Moreover the aperiodic transitions…
