Nature of the Spin Liquid Ground State of the S=1/2 Kagome Heisenberg Model

Depenbrock, Stefan; McCulloch, Ian P.; Schollwoeck, Ulrich

doi:10.48550/arxiv.1205.4858

Cited by 10 publications

(17 citation statements)

References 64 publications

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“…On the theoretical side, they offer a natural framework to investigate and classify the possible phases of quantum matter [68][69][70][71][72] . On the numerical side, they are the basis of novel computational approaches capable of addressing non-perturbatively a large range of interacting systems, including two-dimensional systems of frustrated spins [36][37][38]47,57 and of interacting fermions. [59][60][61][62][63][64][65][66][67] Tensor network states for one dimensional systems include the matrix product state [1][2][3][4] (MPS), which is the basis of the density matrix renormalization group 5,6 (DMRG) algorithm for computing ground states and the time-evolving block-decimation 18 (TEBD) algorithm for simulating time evolution; the tree tensor network 29 (TTN); and the multi-scale entanglement renormalization ansatz 30,31 (MERA) for critical systems.…”

Section: Introductionmentioning

confidence: 99%

“…[59][60][61][62][63][64][65][66][67] Tensor network states for one dimensional systems include the matrix product state [1][2][3][4] (MPS), which is the basis of the density matrix renormalization group 5,6 (DMRG) algorithm for computing ground states and the time-evolving block-decimation 18 (TEBD) algorithm for simulating time evolution; the tree tensor network 29 (TTN); and the multi-scale entanglement renormalization ansatz 30,31 (MERA) for critical systems. In two (and more) spatial dimensions, one can still use a MPS [36][37][38] or TTN, 39,40 although these tensor networks can only represent small systems since they do not offer an efficient, scalable description. In contrast, a projected entangled pair state 48 (PEPS), which is a higher dimensional generalization of MPS, as well as higher dimensional versions of the MERA, 45 offer a scalable description in two and larger dimensions.…”

Section: Introductionmentioning

confidence: 99%

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Tensor network states and algorithms in the presence of a global SU(2) symmetry

2012

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The benefits of exploiting the presence of symmetries in tensor network algorithms have been extensively demonstrated in the context of matrix product states (MPSs). These include the ability to select a specific symmetry sector (e.g. with a given particle number or spin), to ensure the exact preservation of total charge, and to significantly reduce computational costs. Compared to the case of a generic tensor network, the practical implementation of symmetries in the MPS is simplified by the fact that tensors only have three indices (they are trivalent, just as the Clebsch-Gordan coefficients of the symmetry group) and are organized as a one-dimensional array of tensors, without closed loops. Instead, a more complex tensor network, one where tensors have a larger number of indices and/or a more elaborate network structure, requires a more general treatment. In two recent papers, namely (i) [Phys. Rev. A 82, 050301 (2010)] and (ii) [Phys. Rev. B 83, 115125 (2011)], we described how to incorporate a global internal symmetry into a generic tensor network algorithm based on decomposing and manipulating tensors that are invariant under the symmetry. In (i) we considered a generic symmetry group G that is compact, completely reducible and multiplicity free, acting as a global internal symmetry. Then in (ii) we described the implementation of Abelian group symmetries in much more detail, considering a U(1) symmetry (e.g., conservation of global particle number) as a concrete example. In this paper we describe the implementation of non-Abelian group symmetries in great detail. For concreteness we consider an SU(2) symmetry (e.g., conservation of global quantum spin). Our formalism can be readily extended to more exotic symmetries associated with conservation of total fermionic or anyonic charge. As a practical demonstration, we describe the SU(2)-invariant version of the multi-scale entanglement renormalization ansatz and apply it to study the low energy spectrum of a quantum spin chain with a global SU(2) symmetry.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Tensor network states and algorithms in the presence of a global SU(2) symmetry

2012

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“…On the theoretical side, new ways of diagnosing the presence and type of topological order from knowledge of the ground state wave-function alone, based on entanglement entropy [11,12], entanglement spectrum [13] and modular transformations [14,15], have been put forward. On the computational side, the advent of tensor networks [16][17][18][19][20][21][22][23] makes it now possible, by mimicking the structure of entanglement, to efficiently represent a large class of low energy many-body states.…”

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confidence: 99%

“…From an infinite cylinder to a finite torus.-Using an infinite cylinder has the major advantage, compared to previous DMRG studies on a finite cylinder [21][22][23], that it provides a complete set of ground states {|Ψ cyl i } of a local Hamiltonian H. In addition, we can also produce a complete basis {|Ψ tor i } for the (quasi-degenerate) ground space of H on a finite torus of size L x × L y , where the choice L x = L y ensures that also L x ≫ ξ. This is accomplished by reconnecting a region of size L x × L y of the tensor network for {|Ψ cyl i } into a torus, see Fig.…”

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confidence: 99%

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

2013

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Given a microscopic lattice Hamiltonian for a topologically ordered phase, we propose a numerical approach to characterize its emergent anyon model and, in a chiral phase, also its gapless edge theory. First, a tensor network representation of a complete, orthonormal set of ground states on a cylinder of infinite length and finite width is obtained through numerical optimization. Each of these ground states is argued to have a different anyonic flux threading through the cylinder. Then a quasiorthogonal basis on the torus is produced by chopping off and reconnecting the tensor network representation on the cylinder. From these two bases, and by using a number of previous results, most notably the recent proposal of Y. Zhang et al. [Phys. Rev. B 85, 235151 (2012)] to extract the modular U and S matrices, we obtain (i) a complete list of anyon types i, together with (ii) their quantum dimensions d(i) and total quantum dimension D, (iii) their fusion rules N(ij)(k), (iv) their mutual statistics, as encoded in the off-diagonal entries S(ij) of S, (v) their self-statistics or topological spins θ(i), (vi) the topological central charge c of the anyon model, and, in a chiral phase (vii) the low energy spectrum of each sector of the boundary conformal field theory. As a concrete application, we study the hard-core boson Haldane model by using the two-dimensional density matrix renormalization group. A thorough characterization of its universal bulk and edge properties unambiguously shows that it realizes a ν=1/2 bosonic fractional quantum Hall state.

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“…In fact, very recently, frustrated Heisenberg antiferromagnets on kagome and other lattices are reported to be in the RVB spin liquid phase (Z 2 topological phase) by several authors. [12][13][14] Now that the existence of the RVB spin liquid phase appears to be confirmed in QDMs as well as in antiferromagnets, the issue of superconductivity in doped spin liquids becomes a more pressing question. This isssue started in fact to be investigated shortly after the apearence of QDM 11 Doping of an RVB spin liquid is expected to induce a novel type of elementary excitations called holon.…”

Section: Introductionmentioning

confidence: 99%

Hole statistics and superfluid phases in quantum dimer models

et al. 2013

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Quantum dimer models (QDMs) arise as low-energy effective models for frustrated magnets. Some of these models have proven successful in generating a scenario for exotic spin liquid phases with deconfined spinons. Doping, i.e., the introduction of mobile holes, has been considered within the QDM framework and partially studied. A fundamental issue is the possible existence of a superconducting phase in such systems and its properties. For this purpose, the question of the statistics of the mobile holes (or "holons") shall be addressed first. Such issues are studied in detail in this paper for generic doped QDMs defined on the most common two-dimensional lattices (square, triangular, honeycomb, kagome, . . .) and involving general resonant loops. We prove a general "statistical transmutation" symmetry of such doped QDMs by using composite operators of dimers and holes. This exact transformation enables us to define duality equivalence classes (or families) of doped QDMs, and provides the analytic framework to analyze dynamical statistical transmutations. We discuss various possible superconducting phases of the system. In particular, the possibility of an exotic superconducting phase originating from the condensation of (bosonic) charge-e holons is examined. A numerical evidence of such a superconducting phase is presented in the case of the triangular lattice, by introducing a gauge-invariant holon Green's function. We also make the connection with a Bose-Hubbard model on the kagome lattice which gives rise, as an effective model in the limit of strong interactions, to a doped QDM on the triangular lattice.

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Nature of the Spin Liquid Ground State of the S=1/2 Kagome Heisenberg Model

Cited by 10 publications

References 64 publications

Tensor network states and algorithms in the presence of a global SU(2) symmetry

Tensor network states and algorithms in the presence of a global SU(2) symmetry

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

Hole statistics and superfluid phases in quantum dimer models

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