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arXiv:cond-mat/0405034 · Physical Review B

Topological order in Gutzwiller-projected wave functions for quantum antiferromagnets

Tao Li, Hong-Yu Yang

Published May 10, 2007• 10 citations
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Abstract

We study the topological order in a resonating valence bond (RVB) state derived from the Gutzwiller projection of the BCS-like mean-field state. We propose the construction of the topological excitation on the projected RVB state through the Gutzwiller projection of the mean-field state with inserted ${Z}_{2}$ flux tube. We prove that all projected RVB states derived from bipartite effective theories, no matter what the gauge structure in the mean-field ansatz is, are positive definite in the sense of the Marshall sign rule, which provides a universal origin for the absence of topological order in such an RVB state.

DOI
10.1103/physrevb.75.172502

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arXiv:1212.2703•Physical Review B•Sep 25, 2013•6 citations
We study the topological order in a resonating valence bond (RVB) state derived from the Gutzwiller projection of the BCS-like mean-field state. We propose the construction of the topological excitation on the projected RVB state through the Gutzwiller projection of the mean-field state with inserted Z2{Z}_{2}Z2​ flux tube. We prove that all projected RVB states derived from bipartite effective theories, no matter what the gauge structure in the mean-field ansatz is, are positive definite in the sense of the Marshall sign rule, which provides a universal origin for the absence of topological order in such an RVB state.
Gutzwiller-projected (GP) wave functions have been widely used for describing spin-liquid physics in frustrated magnets and in high-temperature superconductors. Such wave functions are known to represent states of the resonating-valence-bond (RVB) type. In the present work I discuss the RVB structure of a GP singlet superconducting state with nodes in the spectrum. The resulting state for the undoped spin system may be described in terms of a ``path integral'' over loop coverings of the lattice, thus extending the known construction for RVB states. The problem of topological order in GP states may be reformulated in terms of the statistical behavior of loops. The simple example of the projected ddd-wave state on the square lattice demonstrates that the statistical behavior of loops is renormalized in a nontrivial manner by the projection.
The nature of the kagome Heisenberg antiferromagnet (HAFM) is under ongoing debate. While recent evidence points towards a Z2{\mathbb{Z}}_{2}Z2​ topological spin liquid, the exact nature of the topological phase is still unclear. In this paper, we introduce semionic resonating valence-bond (RVB) states, this is, resonating valence-bond states which are in the Z2{\mathbb{Z}}_{2}Z2​-ordered double-semion phase, and study them using projected entangled pair states. We investigate their physics and study their suitability as an ansatz for the HAFM, as compared to a conventional RVB state which is in the toric-code Z2{\mathbb{Z}}_{2}Z2​ topological phase. In particular, we find that a suitably optimized ``semionic simplex RVB'' outperforms the equally optimized conventional ``simplex RVB'' state, and that the entanglement spectrum (ES) of the semionic RVB behaves very differently from the ES of the conventional RVB, which suggests using the ES to discriminate the two phases. Finally, we also discuss the possible relevance of space-group symmetry breaking in valence-bond wave functions with double-semion topological order.
We show that there are two types of resonating-valence-bond (RVB) liquid phases present in three-dimensional quantum dimer models, corresponding to the deconfining phases of U(1)U(1)U(1) and Z2{Z}_{2}Z2​ gauge theories in d=3+1.d=3+1.d=3+1. The former is found on the bipartite cubic lattice and is the generalization of the critical point in the square lattice quantum dimer model found originally by Rokhsar and Kivelson. The latter exists on the nonbipartite face-centered cubic lattice and generalizes the RVB phase found earlier by us on the triangular lattice. We discuss the excitation spectrum and the nature of the ordering in both cases. Both phases exhibit gapped spinons. In the U(1)U(1)U(1) case we find a collective, linearly dispersing, transverse excitation, which is the photon of the low-energy Maxwell Lagrangian and we identify the ordering as quantum order in Wen's sense. In the Z2{Z}_{2}Z2​ case all collective excitations are gapped and, as in d=2,d=2,d=2, the low-energy description of this topologically ordered state is the purely topological BF action. As a byproduct of this analysis, we unearth a further gapless excitation, the pi0n, in the square lattice quantum dimer model at its critical point.
The Gutzwiller projection allows the construction of an assortment of variational wave functions for strongly correlated systems. For quantum spin S=1/2S=1/2S=1/2 models, Gutzwiller-projected wave functions have resonating-valence-bond structure and may represent states with fractional quantum numbers for the excitations. Using insights obtained from field-theoretical descriptions of fractionalization in two dimensions, we construct candidate wave functions of fractionalized states by projecting specific superconducting states. We explicitly demonstrate the presence of topological order in these states.
We construct a short-range resonating-valence-bond (RVB) state on the ruby lattice, using projected entangled-pair states with bond dimension D=3D=3D=3. By introducing nonlocal moves to the dimer patterns on the torus, we distinguish four distinct sectors in the space of dimer coverings, which is a signature of the topological nature of the RVB wave function. Furthermore, by calculating the reduced density matrix of a bipartition of the RVB state on an infinite cylinder and exploring its entanglement entropy, we confirm the topological nature of the RVB wave function by obtaining a nonzero topological contribution, $\ensuremath{\gamma}=\ensuremath{-}\mathrm{ln}\phantom{\rule{4pt}{0ex}}2$, consistent with that of a Z2{\mathbb{Z}}_{2}Z2​ topological quantum spin liquid. We also calculate the ground-state energy of the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg model on the ruby lattice and compare it with the RVB energy. Finally, we construct a quantum dimer model for the ruby lattice and discuss it as a possible parent Hamiltonian for the RVB wave function.
The topological entanglement entropy (TEE) of Gutzwiller projected RVB state is studied with Monte Carlo simulation. With tricks to speed up the convergence of TEE, we are able to show that the spin liquid state studied in Zhang et al. [Phys. Rev. B 84, 075128 (2011)] actually does not support Z2{Z}_{2}Z2​ topological order. We find that both an approximate Marshall sign structure and a long-ranged RVB amplitude are responsible for the suppression of the vison gap in this spin liquid state. On the other hand, a TEE of ln2\mathrm{ln}2ln2 is clearly demonstrated for a short range RVB state on triangular lattice proposed in Yunoki et al. [Phys. Rev. B 74, 014408 (2006)]. We find the TEE of the RVB wave function is mainly determined by its phase rather than its amplitude. Our results indicate that the effective field theory misses some key information for the topological property of a RVB state.