Characterizing Topological Order by Studying the Ground States on an Infinite Cylinder

Cincio, Łukasz; Vidal, Guifré

doi:10.1103/physrevlett.110.067208

Cited by 219 publications

(302 citation statements)

References 38 publications

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“…However, these phases do have different topological orders, and we can therefore apply a number of recent developments [47,[99][100][101] which demonstrate how the topological order of a system can be extracted from its entanglement properties.…”

Section: Entanglement Invariants For the Identification Of Fqh Phasesmentioning

confidence: 99%

Competing Abelian and non-Abelian topological orders inν=1/3+1/3quantum Hall bilayers

et al. 2015

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Bilayer quantum Hall systems, realized either in two separated wells or in the lowest two sub-bands of a wide quantum well, provide an experimentally realizable way to tune between competing quantum orders at the same filling fraction. Using newly developed density matrix renormalization group techniques combined with exact diagonalization, we return to the problem of quantum Hall bilayers at filling ν = 1/3 + 1/3. We first consider the Coulomb interaction at bilayer separation d, bilayer tunneling energy ∆SAS, and individual layer width w, where we find a phase diagram which includes three competing Abelian phases: a bilayer-Laughlin phase (two nearly decoupled ν = 1/3 layers); a bilayer-spin singlet phase; and a bilayer-symmetric phase. We also study the order of the transitions between these phases. A variety of non-Abelian phases have also been proposed for these systems. While absent in the simplest phase diagram, by slightly modifying the interlayer repulsion we find a robust non-Abelian phase which we identify as the "interlayer-Pfaffian" phase. In addition to non-Abelian statistics similar to the Moore-Read state, it exhibits a novel form of bilayer-spin charge separation. Our results suggest that ν = 1/3 + 1/3 systems merit further experimental study. CONTENTS

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Section: Entanglement Invariants For the Identification Of Fqh Phasesmentioning

confidence: 99%

Competing Abelian and non-Abelian topological orders inν=1/3+1/3quantum Hall bilayers

et al. 2015

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“…Particularly using the infinite-size density matrix renormalization group (DMRG) method [32] or an infinite projected entangled pair states (PEPS) ansatz [33] the ground states in different topological orders can be obtained on an infinite cylinder and labeled with the anyon type by calculating the modular matrices [32,34,35]. To study both commutation relation fractionalization and quantum number fractionalization, we suggest using a cylinder (4n + 2) unit cells wide.…”

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confidence: 99%

Detecting crystal symmetry fractionalization from the ground state: Application toZ2spin liquids on the kagome lattice

2015

Phys. Rev. B

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In quantum spin liquid states, the fractionalized spinon excitations can carry fractional crystal symmetry quantum numbers, and this symmetry fractionalization distinguishes different symmetry-enriched spin liquid states with identical intrinsic topological order. In this work we propose a simple way to detect signatures of such crystal symmetry fractionalizations from the crystal symmetry representations of the ground state wave function. We demonstrate our method on projected Z 2 spin liquid wave functions on the kagome lattice, and show that it can be used to classify generic wave functions. Particularly our method can be used to distinguish several proposed candidates of Z 2 spin liquid states on the kagome lattice. It is well known that anyons in topologically ordered phases can carry symmetry quantum numbers that are quantized to fractional values. In the celebrated example of fractional quantum Hall states, Laughlin quasiparticles carry fractional charge-the quantum number of the U (1) symmetry [1]. In recent years, great progress has been made in understanding the interplay between symmetry and fractionalization in other topologically ordered states. In particular, topological spin liquids exhibit a more subtle kind of symmetry fractionalization, associated with the crystal symmetry of the underlying lattice instead of internal symmetries [2][3][4]. While some aspects of it have been studied for quite a while, crystal symmetry fractionalization has now received renewed attention, due to an increased interest in the role of crystal symmetry in topological phases of matter. This topic is also becoming timely in view of strong numerical evidence for spin liquids on kagome lattice found in the last few years [5][6][7][8][9]. In order to fully pin down the topological nature of the numerically found spin liquid liquid, the complete pattern of crystal symmetry fractionalization needs to be determined.In this work, we offer our perspective on crystal symmetry fractionalization in Z 2 spin liquids. We find that the nontrivial way that crystal symmetry acts on an individual anyon is directly related to the symmetry representation of the topologically ordered ground states, as labeled by the crystal momentum and parity of many-body wave functions. Given that states with different symmetry labels cannot be adiabatically connected, our finding immediately makes it clear that the classification of spin liquids is refined and enriched by taking into account crystal symmetries [10]. Our theoretical result also provides a straightforward method to classify and detect different spin liquids in numerical studies. As a concrete example, we demonstrate that our method can be used to easily distinguish various Z 2 spin liquids on the kagome lattice [11][12][13].We begin by briefly reviewing what is known about crystal symmetry fractionalization in Z 2 spin liquids, and setting up the terminology for our work. A Z 2 spin liquid [14,15] supports three types of anyon excitations: bosonic spinons, fermionic spinons, and visons. ...

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“…Spontaneous time-reversal symmetry breaking and the emergent loop current.-To investigate the possible spontaneous TRS breaking of the ground states, we turn to larger systems in cylinder geometry and obtain the ground states by implementing a DMRG calculation. Indeed, we obtain two TRS breaking states, jΨ LðRÞ i, by random initializations of wave functions in DMRG simulations [41], which are degenerating in energy as expected (as TRS partners for each other). Here, we label different ground states by their chiral nature, where L (R) stands for "left-hand" ("right-hand") chirality.…”

mentioning

confidence: 99%

“…In order to study the ground state phase diagram in the fV 1 ; V 2 ; V 3 g parameter space, we implement the DMRG algorithm [39,40] combined with ED, both of which have been proven to be powerful and complementary tools for studying realistic models containing arbitrary strong and frustrated interactions [41][42][43][44][45][46][47]. We study large systems up to L y ¼ 6 unit cells and keep up to M ¼ 4800 states to guarantee a good convergence (the discarded truncation error is less than 2 × 10 −6 ).…”

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confidence: 99%

Interaction-Driven Spontaneous Quantum Hall Effect on a Kagome Lattice

et al. 2016

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Topological states of matter have been widely studied as being driven by an external magnetic field, intrinsic spin-orbital coupling, or magnetic doping. Here, we unveil an interaction-driven spontaneous quantum Hall effect (a Chern insulator) emerging in an extended fermion-Hubbard model on a kagome lattice, based on a state-of-the-art density-matrix renormalization group on cylinder geometry and an exact diagonalization in torus geometry. We first demonstrate that the proposed model exhibits an incompressible liquid phase with doublet degenerate ground states as time-reversal partners. The explicit spontaneous time-reversal symmetry breaking is determined by emergent uniform circulating loop currents between nearest neighbors. Importantly, the fingerprint topological nature of the ground state is characterized by quantized Hall conductance. Thus, we identify the liquid phase as a quantum Hall phase, which provides a "proof-of-principle" demonstration of the interaction-driven topological phase in a topologically trivial noninteracting band. DOI: 10.1103/PhysRevLett.117.096402 Introduction.-Topological states of matter have led to an ongoing revolution of our fundamental understanding of quantum materials, as exemplified by the discoveries of the integer quantum Hall (IQH) effect [1,2], the quantum anomalous Hall (QAH) effect [3][4][5], and the quantum spin Hall effect [6][7][8]. Crucially, such topological phases emerge from noninteracting electronic structures with nontrivial topology, where a strong magnetic field or a large spin-orbit coupling is typically necessary. Despite great achievements, most of the materials that have exhibited unique topological properties to date can be understood based on noninteracting physics originating from nontrivial band physics. This limitation has inspired recent enthusiasm in exploring topological phases in "trivial" materials [9][10][11][12][13], without the requirement of a nontrivial invariant encoded in a single-particle wave function or band structure. Finding such novel phases can significantly enrich the class of topological materials and thus is of great importance.The strong correlation between electrons can dramatically change the noninteracting physics and induce spontaneous symmetry breakings. Therefore, many-body interaction is expected to enable topological phases in strongly correlated systems by generating circulating currents and spontaneously breaking time-reversal symmetry (TRS), which act as an effective magnetic field or spin-orbit coupling [12,14,15]. This subject has attracted vigorous research due to support from mean-field studies, followed by low-energy renormalization-group analysis [13,16], and the existence of such topological phases has been suggested for the extended Hubbard model on various lattice models [17][18][19][20][21][22][23][24][25][26][27][28][29][30]. However, unbiased numerical simulations, such as

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Characterizing Topological Order by Studying the Ground States on an Infinite Cylinder

Cited by 219 publications

References 38 publications

Competing Abelian and non-Abelian topological orders inν=1/3+1/3quantum Hall bilayers

Competing Abelian and non-Abelian topological orders inν=1/3+1/3quantum Hall bilayers

Detecting crystal symmetry fractionalization from the ground state: Application toZ2spin liquids on the kagome lattice

Interaction-Driven Spontaneous Quantum Hall Effect on a Kagome Lattice

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