Frustration-free Hamiltonian with Topological Order on Graphs
Pramod Padmanabhan, Jintae Kim, Jung Hoon Han
备注：
The paper was written in a manner making it prone to misinterpretation. These models are inspired by higher gauge theories as elaborated in 1907.01608v2 making them similar to the toric code. The ground states have topological entanglement entropy which is accounted for by anyonic excitations. A more comprehensive version will appear in a fresh submission by one of the authors, PP
It is commonly believed that models defined on a closed one-dimensional manifold cannot give rise to topological order. Here we construct frustration-free Hamiltonians which possess both symmetry protected topological order (SPT) on the open chain {\it and} multiple ground state degeneracy (GSD) that is unrelated to global symmetry breaking on the closed chain. Instead of global symmetry breaking, there exists a {\it local} symmetry operator that commutes with the Hamiltonian and connects the multiple ground states, reminiscent of how the topologically distinct ground states of the toric code are connected by various winding operators. Our model solved on an open chain demonstrates symmetry fractionalization as an indication of SPT order and on a general graph the GSD can be shown to scale with the first Betti number - a topological invariant that counts the number of independent cycles or one dimensional holes of the graph.
We prove that uniformly small short-range perturbations do not close the bulk gap above the ground state of frustration-free quantum spin systems that satisfy a standard local topological quantum order condition. In contrast with earlier results, we do not require a positive lower bound for finite-system Hamiltonians uniform in the system size. To obtain this result, we adapt the Bravyi-Hastings-Michalakis strategy to the GNS representation of the infinite-system ground state.
Ginzburg-Landau theory of continuous phase transitions implicitly assumes that microscopic changes are negligible in determining the thermodynamic properties of the system. In this work we provide an example that clearly contrasts with this assumption. We show that topological frustration can change the nature of a second order quantum phase transition separating two different ordered phases. Even more remarkably, frustration is triggered simply by a suitable choice of boundary conditions in a 1D chain. While with every other BC each of two phases is characterized by its own local order parame…
Recently it was highlighted that one-dimensional antiferromagnetic spin models with frustrated boundary conditions, i.e. periodic boundary conditions in a ring with an odd number of elements, may show very peculiar behavior. Indeed the presence of frustrated boundary conditions can destroy the local magnetic orders presented by the models when different boundary conditions are taken into account and induce novel phase transitions. Motivated by these results, we analyze the effects of the introduction of frustrated boundary conditions on several models supporting (symmetry protected) topologica…
Topological Orders from Reflection Positive Frustration-free Hamiltonians
Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心）；Department of Mathematics, Tsinghua University（清华大学数学系）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）
Spin textures with nontrivial topology, such as vortices and skyrmions, have attracted attention as a source of unconventional magnetic, transport, and optical phenomena. Recently, a new generation of topological spin textures has been extensively studied in itinerant magnets; in contrast to the conventional ones induced, e.g., by the Dzyaloshinskii-Moriya interaction in noncentrosymmetric systems, they are characterized by extremely short magnetic periods and stable even in centrosymmetric systems. Here we review such new types of topological spin textures with particular emphasis on their st…
We prove that the entanglement entropy of the ground state of a locally gapped frustration-free 2D lattice spin system satisfies an area law with respect to a vertical bipartition of the lattice into left and right regions. We first establish that the ground state projector of any locally gapped frustration-free 1D spin system can be approximated to within error $ε$ by a degree $O(\sqrt{n\log(ε^{-1})})$ multivariate polynomial in the interaction terms of the Hamiltonian. This generalizes the optimal bound on the approximate degree of the boolean AND function, which corresponds to the special c…
Topological frustration (or topological mechanics) is the existence of classical zero modes that are robust to many but not all distortions of the Hamiltonian. It arises naturally from locality in systems whose interactions form a set of constraints such as in geometrically frustrated magnets and balls and springs metamaterials. For a magnet whose classical limit exhibits topological frustration, an important question is what happens to this topology when the degrees of freedom are quantized and whether such frustration could lead to exotic quantum phases of matter like a spin liquid. We answe…
Frustration-Free Models for Topological Holography: Fusion Spin Chains as Boundary Algebras
Tsinghua University（清华大学）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
In quantum many-body systems with local interactions, the effects of boundary conditions are considered to be negligible, at least for sufficiently large systems. Here we show an example of the opposite. We consider a spin chain with two competing interactions, set on a ring with an odd number of sites. When only the dominant interaction is antiferromagnetic, and thus induces topological frustration, the standard antiferromagnetic order (expressed by the magnetization) is destroyed. When also the second interaction turns from ferro to antiferro, an antiferromagnetic order characterized by a si…
We show that there is an emergent lattice description for the continuous fractional quantum Hall (FQH) systems, with a generalised set of few-body coherent states. In particular, model Hamiltonians of the FQH effect are equivalent to the real space von Neumann lattice of local projection operators imposed on a continuous system in the thermodynamic limit. It can be analytically derived that tuning local one-body potentials in such lattices amounts to the tuning of individual two or few-body pseudopotentials. For some cases, we can realise pure few-body pseudopotentials important for stabilisin…
Ground State Properties of the Doped Kitaev-Heisenberg Chain: Topological Superconducting and Mott Insulating Phases Driven by Magnetic Frustration
University of Warsaw（华沙大学）；TU Wien（维也纳工业大学）；Technical University Dresden（德累斯顿工业大学）；Institute for Theoretical Solid State Physics, IFW Dresden（德累斯顿联邦材料研究所理论固态物理研究所）
We study frustration-free Hamiltonians of fractional quantum Hall (FQH) states from the point of view of the matrix product state (MPS) representation of their ground and excited states. There is a wealth of solvable models relating to FQH physics, which, however, is mostly derived and analyzed from the vantage point of first-quantized "analytic clustering properties". In contrast, one obtains long-ranged frustration-free lattice models when these Hamiltonians are studied in an orbital basis, which is the natural basis for the MPS representation of FQH states. The connection between MPS-like s…
We analyze a system in which a topological Majorana zero mode (tMZM) combines with a Majorana produced by quantum frustration (fMZM) to produce a novel ground state. The system that we study combines two parts, a grounded topological superconducting wire that hosts two tMZMs at its ends, and an on-resonant quantum dot connected to two dissipative leads. The quantum dot with dissipative leads creates an effective two-channel Kondo (2CK) state in which quantum frustration yields an isolated fMZM at the dot. We find that coupling the dot to one end of the topological wire stabilizes the tMZM at t…
Chiral magnetic phases with an unconventional topological twist in the magnetization are of huge interest due to their potential in spintronics applications. Here, we present a general method to induce such exotic magnetic phases using interfacial phase frustration within artificially grown superlattices. To demonstrate our method, we consider a multilayer with two different chiral magnetic phases as the competing orders at the top and bottom and show, using Monte Carlo calculations, that the interfacial phase frustration is realized at the central layer. In particular, we obtain three unconve…
Frustration-driven topological textures on the honeycomb lattice: antiferromagnetic meron-antimeron and skyrmion crystals emerging from spiral spin liquids
We systematically derive low-energy effective Hamiltonians for molecular solids $β^\prime$-$X$[Pd(dmit)$_{2}$]$_{2}$ ($X$ represents a cation) using ab initio density functional theory calculations and clarify how the cation controls the inter-dimer transfer integrals and the interaction parameters. The effective models are solved using the exact diagonalization method and the antiferromagnetic ordered moment is shown to be significantly suppressed around the spin-liquid candidate of $X$=EtMe$_{3}$Sb, which is reported in experiments. We also show that both the geometrical frustration and the …
Institute for Solid State Physics, University of Tokyo（东京大学固体物理研究所）；Department of Physics, Hong Kong University of Science and Technology（香港科技大学物理系）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理化学研究所跨学科理论与数学科学中心（iTHEMS））；Department of Applied Physics, The University of Tokyo（东京大学应用物理系）；Center for Theoretical Condensed Matter Physics, Hong Kong University of Science and Technology（香港科技大学理论凝聚态物理中心）
Spin Frustration and Orbital Order in Vanadium Spinels
本文介绍了钒尖晶石氧化物$A$ V $_2$ O $_4$（$A$ = Zn、Mg或Cd）中的几何阻挫效应和自旋与轨道自由度之间的相互作用的理论研究结果。在强关联极限下引入一个有效的自旋-轨道-晶格耦合模型，并对该模型进行蒙特卡罗模拟，提出了一个轨道有序相的约化自旋哈密顿量，以捕捉反铁磁有序的稳定机制.轨道序通过在自旋交换相互作用中引入空间各向异性大大降低了自旋阻挫，约化自旋模型可以看作是弱耦合的一维反铁磁链.通过有限尺寸标度分析得到的临界指数表明，磁跃迁属于三维海森堡普适类。平均场能级中剩余的挫折通过热涨落来减少，以稳定共线有序。
The University of Tokyo（东京大学）；Institute for Solid State Physics, University of Tokyo（东京大学固体物理研究所）；Hong Kong University of Science and Technology（香港科技大学）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理化学研究所跨学科理论与数学科学中心（iTHEMS））
We study the stability with respect to a broad class of perturbations of gapped ground state phases of quantum spin systems defined by frustration-free Hamiltonians. The core result of this work is a proof using the Bravyi-Hastings-Michalakis (BHM) strategy that under a condition of Local Topological Quantum Order, the bulk gap is stable under perturbations that decay at long distances faster than a stretched exponential. Compared to previous work we expand the class of frustration-free quantum spin models that can be handled to include models with more general boundary conditions, and models …
