Feasibility of self-correcting quantum memory and thermal stability of topological order

Recently, it has become apparent that the thermal stability of topologically ordered systems at finite temperature, as discussed in condensed matter physics, can be studied by addressing the feasibility of self-correcting quantum memory, as discussed in quantum information science. Here, with this correspondence in mind, we propose a model of quantum codes that may cover a large class of physically realizable quantum memory. The model is supported by a certain class of gapped spin Hamiltonians, called stabilizer Hamiltonians, with translation symmetries and a small number of ground states that does not grow with the system size. We show that the model does not work as self-correcting quantum memory due to a certain topological constraint on geometric shapes of its logical operators. This quantum coding theoretical result implies that systems covered or approximated by the model cannot have thermally stable topological order, meaning that systems cannot be stable against both thermal fluctuations and local perturbations simultaneously in two and three spatial dimensions.     

Observation of Topological Links Associated with Hopf Insulators in a Solid-State Quantum Simulator

Hopf insulators are intriguing three-dimensional topological insulators characterized by an integer topological invariant. They originate from the mathematical theory of Hopf fibration and epitomize the deep connection between knot theory and topological phases of matter, which distinguishes them from other classes of topological insulators. Here, we implement a model Hamiltonian for Hopf insulators in a solid-state quantum simulator and report the first experimental observation of their topological properties,including nontrivial topological links associated with the Hopf fibration and the integer-valued topological invariant obtained from a direct tomographic measurement. Our observation of topological links and Hopf fibration in a quantum simulator opens the door to probe rich topological properties of Hopf insulators in experiments. The quantum simulation and probing methods are also applicable to the study of other intricate three-dimensional topological model Hamiltonians.     

PEPS as ground states: Degeneracy and topology

We introduce a framework for characterizing Matrix Product States (MPS) and Projected Entangled Pair States (PEPS) in terms of symmetries. This allows us to understand how PEPS appear as ground states of local Hamiltonians with finitely degenerate ground states and to characterize the ground state subspace. Subsequently, we apply our framework to show how the topological properties of these ground states can be explained solely from the symmetry: We prove that ground states are locally indistinguishable and can be transformed into each other by acting on a restricted region, we explain the origin of the topological entropy, and we discuss how to renormalize these states based on their symmetries. Finally, we show how the anyonic character of excitations can be understood as a consequence of the underlying symmetries.     

Topological invariant for superfluid <Superscript>3</Superscript>He-B and quantum phase transitions

We consider topological invariant describing the vacuum states of superfluid ³He-B, which belongs to the special class of time-reversal invariant topological insulators and superfluids. Discrete symmetries important for classification of the topologically distinct vacuum states are discussed. One of them leads to the additional subclasses of ³He-B states and is responsible for the finite density of states of Majorana fermions living at some interfaces between the bulk states. Integer valued topological invariant is expressed in terms of the Green’s function, which allows us to consider systems with interaction.     

Fragile Mott insulators

We prove that there exists a class of crystalline insulators, which we call "fragile Mott insulators," which are not adiabatically connected to any sort of band insulator provided time-reversal and certain point-group symmetries are respected, but which are otherwise unspectacular in that they exhibit no topological order nor any form of fractionalized quasiparticles. Different fragile Mott insulators are characterized by different nontrivial one-dimensional representations of the crystal point group. We illustrate this new type of insulators with two examples: the d Mott insulator discovered in the checkerboard Hubbard model at half-filling and the Affleck-Kennedy-Lieb-Tasaki insulator on the square lattice.     

Bott Periodicity for {\mathbb{Z}_2} Symmetric Ground States of Gapped Free-Fermion Systems

Building on the symmetry classification of disordered fermions, we give a proof of the proposal by Kitaev, and others, for a “Bott clock” topological classification of free-fermion ground states of gapped systems with symmetries. Our approach differs from previous ones in that (i) we work in the standard framework of Hermitian quantum mechanics over the complex numbers, (ii) we directly formulate a mathematical model for ground states rather than spectrally flattened Hamiltonians, and (iii) we use homotopy-theoretic tools rather than K-theory. Key to our proof is a natural transformation that squares to the standard Bott map and relates the ground state of a d-dimensional system in symmetry class s to the ground state of a (d + 1)-dimensional system in symmetry class s + 1. This relation gives a new vantage point on topological insulators and superconductors.     

Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes

Searches for possible new quantum phases and classifications of quantum phases have been central problems in physics. Yet, they are indeed challenging problems due to the computational difficulties in analyzing quantum many-body systems and the lack of a general framework for classifications. While frustration-free Hamiltonians, which appear as fixed point Hamiltonians of renormalization group transformations, may serve as representatives of quantum phases, it is still difficult to analyze and classify quantum phases of arbitrary frustration-free Hamiltonians exhaustively. Here, we address these problems by sharpening our considerations to a certain subclass of frustration-free Hamiltonians, called stabilizer Hamiltonians, which have been actively studied in quantum information science. We propose a model of frustration-free Hamiltonians which covers a large class of physically realistic stabilizer Hamiltonians, constrained to only three physical conditions; the locality of interaction terms, translation symmetries and scale symmetries, meaning that the number of ground states does not grow with the system size. We show that quantum phases arising in two-dimensional models can be classified exactly through certain quantum coding theoretical operators, called logical operators, by proving that two models with topologically distinct shapes of logical operators are always separated by quantum phase transitions.     

Robust preparation and manipulation of protected qubits using time-varying Hamiltonians

We show that it is possible to initialize and manipulate in a deterministic manner protected qubits using time-varying Hamiltonians. Taking advantage of the symmetries of the system, we predict the effect of the noise during the initialization and manipulation. These predictions are in good agreement with numerical simulations. Our study shows that the topological protection remains efficient under realistic experimental conditions.     

Topological spin Hall states, charged skyrmions, and superconductivity in two dimensions

We study the properties of two dimensional topological spin Hall insulators which arise through spontaneous breakdown of spin symmetry in systems that are spin rotation invariant. Such a phase breaks spin rotation but not time reversal symmetry and has a vector order parameter. Skyrmion configurations in this vector order parameter are shown to have an electric charge that is twice the electron charge. When the spin Hall order is destroyed by condensation of Skyrmions superconductivity results. This may happen either through doping or at fixed filling by tuning interactions to close the Skyrmion gap. In the latter case the superconductor-spin Hall insulator quantum phase transition can be second order even though the two phases break distinct symmetries.     

A symmetry principle for topological quantum order

We present a unifying framework to study physical systems which exhibit topological quantum order (TQO). The major guiding principle behind our approach is that of symmetries and entanglement. These symmetries may be actual symmetries of the Hamiltonian characterizing the system, or emergent symmetries. To this end, we introduce the concept of low-dimensional Gauge-like symmetries (GLSs), and the physical conservation laws (including topological terms, fractionalization, and the absence of quasi-particle excitations) which emerge from them. We prove then sufficient conditions for TQO at both zero and finite temperatures. The physical engine for TQO are topological defects associated with the restoration of GLSs. These defects propagate freely through the system and enforce TQO. Our results are strongest for gapped systems with continuous GLSs. At zero temperature, selection rules associated with the GLSs enable us to systematically construct general states with TQO; these selection rules do not rely on the existence of a finite gap between the ground states to all other excited states. Indices associated with these symmetries correspond to different topological sectors. All currently known examples of TQO display GLSs. Other systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin-exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. The symmetry based framework discussed herein allows us to go beyond standard topological field theories and systematically engineer new physical models with finite temperature TQO (both Abelian and non-Abelian). Furthermore, we analyze the insufficiency of entanglement entropy (we introduce SU(N) Klein models on small world networks to make the argument even sharper), spectral structures, maximal string correlators, and fractionalization in establishing TQO. We show that Kitaev’s Toric code model and Wen’s plaquette model are equivalent and reduce, by a duality mapping, to an Ising chain, demonstrating that despite the spectral gap in these systems the toric operator expectation values may vanish once thermal fluctuations are present. This illustrates the fact that the quantum states themselves in a particular (operator language) representation encode TQO and that the duality mappings, being non-local in the original representation, disentangle the order. We present a general algorithm for the construction of long-range string and brane orders in general systems with entangled ground states; this algorithm relies on general ground states selection rules and becomes of the broadest applicability in gapped systems in arbitrary dimensions. We exactly recast some known non-local string correlators in terms of local correlation functions. We discuss relations to problems in graph theory.     

Novel topological insulators from crystalline symmetries

We discuss recent advances in the study of topological insulators protected by spatial symmetries by reviewing three representative, theoretical examples. In three dimensions (3D), these states of matter are generally characterized by the presence of gapless boundary states at surfaces that respect the protecting spatial symmetry. We discuss the appearance of these topological states in both crystals with negligible spin–orbit coupling and a fourfold rotational symmetry, as well as in mirror-symmetric crystals with sizable spin–orbit interaction characterized by the so-called mirror Chern number. Finally, we also discuss similar topological crystalline states in one-dimensional (1D) insulators, such as nanowires or atomic chains, with mirror symmetry. There, the prime physical consequence of the non-trivial topology is the presence of quantized end charges.     

Topological order parameters for interacting topological insulators

We propose a topological order parameter for interacting topological insulators, expressed in terms of the full Green's functions of the interacting system. We show that it is exactly quantized for a time-reversal invariant topological insulator, and it can be experimentally measured through the topological magneto-electric effect. This topological order parameter can be applied to both interacting and disordered systems, and used for determining their phase diagrams.     

Z_2拓扑不变量与拓扑绝缘体

文章回顾了几种Z2拓扑数的计算方法,并详细介绍了一种用非阿贝尔贝里联络表示绝缘体Z2不变量的计算方法.这种方法可以确定出一般能带绝缘体的拓扑性质,而不需要限定波函数的规范.利用这种新方法,文章作者计算了二维石墨烯(graphene)系统的Z2拓扑数,得到了和以前研究相一致的结论.     

Tunable topological quantum states in three- and two-dimensional materials

We review our theoretical advances in tunable topological quantum states in three- and twodimensional materials with strong spin–orbital couplings. In three-dimensional systems, we propose a new tunable topological insulator, bismuth-based skutterudites in which topological insulating states can be induced by external strains. The orbitals involved in the topological band-inversion process are the d- and p-orbitals, unlike typical topological insulators such as Bi₂Se₃and BiTeI, where only the p-orbitals are involved in the band-inversion process. Owing to the presence of large d-electronic states, the electronic interaction in our proposed topological insulator is much stronger than that in other conventional topological insulators. In two-dimensional systems, we investigated 3d-transition-metal-doped silicene. Using both an analytical model and first-principles Wannier interpolation, we demonstrate that silicene decorated with certain 3d transition metals such as vanadium can sustain a stable quantum anomalous Hall effect. We also predict that the quantum valley Hall effect and electrically tunable topological states could be realized in certain transition-metal-doped silicenes where the energy band inversion occurs. These findings provide realistic materials in which topological states could be arbitrarily controlled.     

Dynamical symmetries and Nambu mechanics

It is shown that several Hamiltonian systems possessing dynamical or hidden symmetries can be realized within the framework of Nambu's generalized mechanics. Among such systems are the SU(n)-isotropic harmonic oscillator and the SO(4) Kepler problem. As required by the formulation of Nambu dynamics, the integrals of motion for these systems necessarily become the so-called generalized Hamiltonians. Furthermore, in most of these problems, the definition of these generalized Hamiltonians is not unique.     

Non-Hermitian topological Anderson insulators

Non-Hermitian systems can exhibit exotic topological and localization properties.Here we elucidate the non-Hermitian effects on disordered topological systems using a nonreciprocal disordered Su-Schrieffer-Heeger model.We show that the non-Hermiticity can enhance the topological phase against disorders by increasing bulk gaps.Moreover,we uncover a topological phase which emerges under both moderate non-Hermiticity and disorders,and is characterized by localized insulating bulk states with a disorder-averaged winding number and zero-energy edge modes.Such topological phases induced by the combination of non-Hermiticity and disorders are dubbed non-Hermitian topological Anderson insulators.We reveal that the system has unique non-monotonous localization behavior and the topological transition is accompanied by an Anderson transition.These properties are general in other non-Hermitian models.     

Symmetry and Topology of the Topological Counterparts in Non-Hermitian Systems

The symmetries and topological properties of the topological counterparts in 1D non-Hermitian systems are investigated. It is found that, after applying the non-unitary similarity transformation, the non-unitary topological counterpart in real space exhibits completely different global symmetries except for the sublattice symmetry and reveals many brand new local symmetries. Due to the abundant symmetries of non-unitary topological counterparts, it is also found that the unique overlapping projections about the unit sphere vector representing the eigenstates appear in the nontrivial regions, and the triviality of the point-gap topology of non-unitary topological counterpart completely eliminate the intrinsic skin effect in non-Hermitian systems. It is also shown that the unitary topological counterpart never arises any changes for the original symmetries and topological structures even in real space. Unitary topological counterparts are further summarized about the two-band Bloch Hamiltonian, which can expand the definition of non-Bloch winding number. Furthermore, it is demonstrated theoretically that the Bloch Hamiltonian would still hold time-reversal symmetry, abnormal particle-hole symmetry, and sublattice symmetry even suffering from the non-unitary transformation. This work provides a new way to understand the roles of symmetry and topology in non-Hermitian systems from the perspective of topological counterparts.     

Hamilton系统Noether理论的新型逆问题

研究Hamilton系统Noether理论新型的逆问题,得到利用Noether理论从已知的第一积分构建Hamilton函数和对称性的一般解法和若干特殊解法,提出由Hamilton函数直接导出守恒量的两条推论.举例说明所得结果的应用.     

动力学淬火过程中的不动点及衍生拓扑现象

本文对近两年来有关淬火动力学过程中拓扑现象的研究做简要综述.这些动力学拓扑现象被动力学过程中的衍生拓扑不变量保护,与淬火前后体系的拓扑性质有密切关系.基于人工量子模拟平台的高度可控性,已在诸如超冷原子、超导量子比特、核磁共振、线性光学等众多物理体系中,通过对人工拓扑体系动力学过程的调控,观测到如动力学涡旋、动量-时间域的Hopf映射及环绕数、拓扑保护的自旋环结构、动力学量子相变、动量-时间斯格明子等诸多动力学拓扑现象.其中某些拓扑结构还可以在非幺正动力学淬火过程中稳定存在.这些研究将人们对拓扑物相的认识和研究从平衡态推广到非平衡动力学领域,具有重要的科学价值.