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arXiv:cond-mat/0509791 · Physical Review B

Resonating-valence-bond structure of Gutzwiller-projected superconducting wave functions

Dmitri A. Ivanov

Published Jul 27, 2006• 6 citations
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Abstract

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 $d$-wave state on the square lattice demonstrates that the statistical behavior of loops is renormalized in a nontrivial manner by the projection.

DOI
10.1103/physrevb.74.024525

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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.
We construct a family of simple fermionic projected entangled pair states (fPEPS) on the square lattice with bond dimension D=3D=3D=3 which are exactly hole-doped resonating valence bond (RVB) wave functions with short-range singlet bonds. Under doping the insulating RVB spin liquid evolves immediately into a superconductor with mixed d+isd+isd+is pairing symmetry whose pair amplitude grows as the square root of the doping. The relative weight between sss-wave and ddd-wave components can be controlled by a single variational parameter ccc. We optimize our ansatz with respect to ccc for the frustrated $t\text{\ensuremath{-}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ model (including both nearest and next-nearest neighbor antiferromagnetic interactions J1{J}_{1}J1​ and J2{J}_{2}J2​, respectively) for ${J}_{2}\ensuremath{\simeq}{J}_{1}/2$ and obtain an energy very close to the infinite-PEPS state (using full update optimization and same bond dimension). The orbital symmetry of the optimized RVB superconductor has predominant ddd-wave character, although we argue a residual (complex sss-wave) time reversal symmetry breaking component should always be present. Connections of the results to the physics of superconducting cuprates and pnictides are outlined.
We review the resonating valence bond (RVB) theory of high-temperature superconductivity using Gutzwiller projected wave functions that incorporate strong correlations. After a general overview of the phenomenon of high-temperature superconductivity, we discuss Anderson's RVB picture and its implementation by renormalized mean-field theory (RMFT) and variational Monte Carlo (VMC) techniques. We review RMFT and VMC results with an emphasis on recent developments in extending VMC and RMFT techniques to excited states. We compare results obtained from these methods with angle-resolved photoemission spectroscopy (ARPES) and scanning tunnelling microscopy (STM). We conclude by summarizing recent successes of this approach and discuss open problems that need to be solved for a consistent and complete description of high-temperature superconductivity using Gutzwiller projected wave functions.
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.
We review the Resonating Valence Bond (RVB) theory of high temperatur e superconductivity using Gutzwiller projected wave functions that incorporate strong correlations. After a general overview of the phenomenon of high temperature superconductivity, we discuss Anderson's RVB picture and its implementation by renormalised mean field theory (RMFT) and variational Monte Carlo (VMC) techniques. We review RMFT and VMC results with an emphasis on recent development s in extending VMC and RMFT techniques to excited states. We compare results obtained from these methods with angle resolved photoemission spectroscopy (ARPES) and scanning tunnelling microscopy (STM). We conclude by summarising recent successes of this approach and discuss open problems that need to be solved for a consistent and complete description of high temperature superconductivity using Gutzwiller projected wave functions.
Twenty-five years after the first proposal, the question whether the ground state of a frustrated spin-half system is well described by a spin-liquid Resonating Valence Bond (RVB) wave function is still controversial. A physically transparent representation of a RVB state can be obtained in fermionic representation with a standard BCS-type pairing wave function, working in the subspace with fixed number of electrons and no double occupancies. In this work, we show that, using this variational wave function with a careful parameterization of the pairing function, it is possible to obtain an extremely accurate {\em ansatz} for the ground state of the Heisenberg antiferromagnet with next-nearest neighbors interactions (J1−J2J_{1}{-}J_{2}J1​−J2​ model) in the regime of strong frustration. Indeed, in the spin-half realization of this model, it is known that the combined effect of frustration and zero-point motion interferes with the mechanism of spontaneously broken symmetry, giving rise to a non-magnetic phase of purely quantum-mechanical nature (J2/J1≃0.5J_2/J_1\simeq 0.5J2​/J1​≃0.5). This wave function is proposed to represent the generic spin-half RVB ground state in spin liquids.
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.