Many-body generalization of the Z2 topological invariant for the quantum spin Hall effect

We propose a many-body generalization of the Z2 topological invariant for the quantum spin Hall insulator, which does not rely on single-particle band structures. The invariant is derived as a topological obstruction that distinguishes topologically distinct many-body ground states on a torus. It is also expressed as a Wilson loop of the SU(2) Berry gauge field, which is quantized due to time-reversal symmetry.     

Lieb–Schultz–Mattis Theorem with a Local Twist for General One-Dimensional Quantum Systems

We formulate and prove the local twist version of the Yamanaka–Oshikawa–Affleck theorem, an extension of the Lieb–Schultz–Mattis theorem, for one-dimensional systems of quantum particles or spins. We can treat almost any translationally invariant system with global U(1) symmetry. Time-reversal or inversion symmetry is not assumed. It is proved that, when the “filling factor” is not an integer, a ground state without any long-range order must be accompanied by low-lying excitations whose number grows indefinitely as the system size is increased. The result is closely related to the absence of topological order in one-dimension. The present paper is written in a self-contained manner, and does not require any knowledge of the Lieb–Schultz–Mattis and related theorems.     

Energy landscape of 3D spin Hamiltonians with topological order

We explore the feasibility of a quantum self-correcting memory based on 3D spin Hamiltonians with topological quantum order in which thermal diffusion of topological defects is suppressed by macroscopic energy barriers. To this end we characterize the energy landscape of stabilizer code Hamiltonians with local bounded-strength interactions which have a topologically ordered ground state but do not have stringlike logical operators. We prove that any sequence of local errors mapping a ground state of such a Hamiltonian to an orthogonal ground state must cross an energy barrier growing at least as a logarithm of the lattice size. Our bound on the energy barrier is tight up to a constant factor for one particular 3D spin Hamiltonian.     

Class of quantum many-body states that can be efficiently simulated

We introduce the multiscale entanglement renormalization ansatz, a class of quantum many-body states on a D-dimensional lattice that can be efficiently simulated with a classical computer, in that the expectation value of local observables can be computed exactly and efficiently. The multiscale entanglement renormalization ansatz is equivalent to a quantum circuit of logarithmic depth that has a very characteristic causal structure. It is also the ansatz underlying entanglement renormalization, a novel coarse-graining scheme for many-body quantum systems on a lattice.     

Topological order at nonzero temperature

We propose a definition for topological order at nonzero temperature in analogy to the usual zero temperature definition that a state is topologically ordered, or "nontrivial", if it cannot be transformed into a product state (or a state close to a product state) using a local (or approximately local) quantum circuit. We prove that any two-dimensional Hamiltonian which is a sum of commuting local terms is not topologically ordered at T > 0. We show that such trivial states cannot be used to store quantum information using certain stringlike operators. This definition is not too restrictive, however, as the four dimensional toric code does have a nontrivial phase at nonzero temperature.     

利用超导量子电路模拟拓扑量子材料

近年来,探索新的拓扑量子材料、研究拓扑材料的新奇物理性质成为凝聚态物理领域的一个热点.但是,由于合成、测量等手段的限制,人们难以在真实材料中实现和观测很多理论预言的材料及其物理性质,促使量子模拟日益成为研究量子多体系统的一个重要手段.作为全固态器件,超导量子电路是一个在扩展性、集成性、调控性上都具有巨大优势的人工量子系统,是实现量子模拟的重要方案.本文总结了利用超导量子电路对时间-空间反演对称性保护的拓扑半金属、Hopf-link半金属和Maxwell半金属等拓扑材料的量子模拟,显示出超导量子电路在模拟凝聚态物理系统方面具有广阔前景.     

Criticality, the area law, and the computational power of projected entangled pair states

The projected entangled pair state (PEPS) representation of quantum states on two-dimensional lattices induces an entanglement based hierarchy in state space. We show that the lowest levels of this hierarchy exhibit a very rich structure including states with critical and topological properties. We prove, in particular, that coherent versions of thermal states of any local 2D classical spin model correspond to such PEPS, which are in turn ground states of local 2D quantum Hamiltonians. This correspondence maps thermal onto quantum fluctuations, and it allows us to analytically construct critical quantum models exhibiting a strict area law scaling of the entanglement entropy in the face of power law decaying correlations. Moreover, it enables us to show that there exist PEPS which can serve as computational resources for the solution of NP-hard problems.     

The insulating state of matter: a geometrical theory

In 1964 Kohn published the milestone paper “Theory of the insulating state”, according to which insulators and metals differ in their ground state. Even before the system is excited by any probe, a different organization of the electrons is present in the ground state and this is the key feature discriminating between insulators and metals. However, the theory of the insulating state remained somewhat incomplete until the late 1990s; this review addresses the recent developments. The many-body ground wavefunction of any insulator is characterized by means of geometrical concepts (Berry phase, connection, curvature, Chern number, quantum metric). Among them, it is the quantum metric which sharply characterizes the insulating state of matter. The theory deals on a common ground with several kinds of insulators: band insulators, Mott insulators, Anderson insulators, quantum Hall insulators, Chern and topological insulators.     

Ground states of Fermions on lattices

We consider Fermion systems on integer lattices. We establish the existence of dynamics for a class of long range interactions. The infinite volume ground states are considered. The equivalence of the variational principle and ground state conditions is proved for long range interactions. We also prove that any pure translationally invariant ground state of the gauge invariant algebra is extendible to a ground state of the full CAR algebra for the Hamiltonian with a chemical potential (equivalence of ensemble for canonical and ground canonical states at the zero temperature).     

A two-dimensional isotropic quantum antiferromagnet with unique disordered ground state

We continue the study of valence-bond solid antiferromagnetic quantum Hamiltonians. These Hamiltonians are invariant under rotations in spin space. We prove that a particular two-dimensional model from this class (the spin-3/2 model on the hexagonal lattice) has a unique ground state in the infinite-volume limit and hence no Néel order. Moreover, all truncated correlation functions decay exponentially in this ground state. We also characterize all the finite-volume ground states of these models (in every dimension), and prove that the two-point correlation function of the spin-2 square lattice model with periodic boundary conditions has exponential decay.     

Structure of 2D Topological Stabilizer Codes

We provide a detailed study of the general structure of translationally invariant two-dimensional topological stabilizer quantum error correcting codes, including subsystem codes. We show that they can be understood in terms of the homology of string operators that carry a certain topological charge. In subsystem codes, two dual kinds of charges appear. We prove that two non-chiral codes are equivalent under local transformations iff they have isomorphic topological charges. Our approach emphasizes local properties over global ones.     

基于张量网络算法的自旋梯子系统的弦序参量的研究

通过基于矩阵乘积态(MPS)的强关联电子量子自旋梯子格点系统的张量网络(TN)算法,摸索研究自旋梯子量子多体系统的弦序参量,探测系统的量子相变点,刻画系统的量子临界现象,获取系统的量子相图,这为我们提供了一个研究自旋梯子系统的量子多体物理性质强有力的工具和方法:在不知道系统是否缺乏Landau对称性破缺序或者系统是否存在相关的拓扑弦序的情况下,可以先得到系统的基态波函数,如果基态缺乏Landau对称性破缺序,或可以通过其它方式找出系统存在若干非局域的弦序参量,来完整地描述一些拓扑量子相变点,获得系统的量子相图,从而丰富和发展了传统的Landau对称性破缺的相变理论.     

Ground states of quantum spin systems

We prove that ground states of quantum spin systems are characterized by a principle of minimum local energy and that translationally invariant ground states are characterized by the principle of minimum energy per unit volume.     

Quantum spin chains with quantum group symmetry

We consider actions of quantum groups on lattice spin systems. We show that if an action of a quantum group respects the local structure of a lattice system, it has to be an ordinary group. Even allowing weakly delocalized (quasi-local) tails of the action, we find that there are no actions of a properly quantum group commuting with lattice translations. The non-locality arises from the ordering of factors in the quantum groupC ^*-algebra, and can be made one-sided, thus allowing semi-local actions on a half chain. Under such actions, localized quantum group invariant elements remain localized. Hence the notion of interactions invariant under the quantum group and also under translations, recently studied by many authors, makes sense even though there is no global action of the quantum group. We consider a class of such quantum group invariant interactions with the property that there is a unique translation invariant ground state. Under weak locality assumptions, its GNS representation carries no unitary representation of the quantum group.Supported in part by NSF Grant # PHY90-19433 A02Copyright © 1995 by the authors. Faithful reproduction of this article by any means is permitted for non-commercial purposes.     

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.     

Double Semions in Arbitrary Dimension

We present a generalization of the double semion topological quantum field theory to higher dimensions, as a theory of \({d-1}\) dimensional surfaces in a d dimensional ambient space. We construct a local Hamiltonian that is a sum of commuting projectors and analyze the excitations and the ground state degeneracy. Defining a consistent set of local rules requires the sign structure of the ground state wavefunction to depend not just on the number of disconnected surfaces, but also upon their higher Betti numbers through the semicharacteristic. For odd d the theory is related to the toric code by a local unitary transformation, but for even d the dimension of the space of zero energy ground states is in general different from the toric code and for even \({d > 2}\) it is also in general different from that of the twisted \({Z_2}\) Dijkgraaf–Witten model.     

微波调制的里德堡原子集体量子跳跃

对于一个三能级原子体系,原子的两个基态能级通过微波耦合起来,其中一个基态能级可被激发到里德堡态,从而可观察量子跳跃现象.本文采用量子轨线方法研究了微波调制的里德堡原子集体量子跳跃.研究结果表明,微波耦合基态能级可以提高光子关联,增强光子聚束效应,即使较少的原子中也可以观察到系统在高里德堡占据数态和低里德堡占据数态之间的切换.这一结果为将来进一步研究里德堡自旋晶格中的多体动力学提供了新思路.     

Topological order and semions in a strongly correlated quantum spin Hall insulator

We provide a self-consistent mean-field framework to study the effect of strong interactions in a quantum spin Hall insulator on the honeycomb lattice. We identify an exotic phase for large spin-orbit coupling and intermediate Hubbard interaction. This phase is gapped and does not break any symmetry. Instead, we find a fourfold topological degeneracy of the ground state on the torus and fractionalized excitations with semionic mutual braiding statistics. Moreover, we argue that it has gapless edge modes protected by time-reversal symmetry but a trivial Z(2) topological invariant. Finally, we discuss the experimental signatures of this exotic phase. Our work highlights the important theme that interesting phases arise in the regime of strong spin-orbit coupling and interactions.     

Quantum glassiness in strongly correlated clean systems: an example of topological overprotection

This Letter presents solvable examples of quantum many-body Hamiltonians of systems that are unable to reach their ground states as the environment temperature is lowered to absolute zero. These examples, three-dimensional generalizations of quantum Hamiltonians proposed for topological quantum computing, (1) have no quenched disorder, (2) have solely local interactions, (3) have an exactly solvable spectrum, (4) have topologically ordered ground states, and (5) have slow dynamical relaxation rates akin to those of strong structural glasses.     

{\mathbb{Z}_{2}} Invariants of Topological Insulators as Geometric Obstructions

We consider a gapped periodic quantum system with time-reversal symmetry of fermionic (or odd) type, i.e. the time-reversal operator squares to \({-\mathbb{1}}\). We investigate the existence of periodic and time-reversal invariant Bloch frames in dimensions 2 and 3. In 2d, the obstruction to the existence of such a frame is shown to be encoded in a \({\mathbb{Z}_2}\)-valued topological invariant, which can be computed by a simple algorithm. We prove that the latter agrees with the Fu-Kane index. In 3d, instead, four \({\mathbb{Z}_2}\) invariants emerge from the construction, again related to the Fu-Kane-Mele indices. When no topological obstruction is present, we provide a constructive algorithm yielding explicitly a periodic and time-reversal invariant Bloch frame. The result is formulated in an abstract setting, so that it applies both to discrete models and to continuous ones.