Gutzwiller–RVB theory of high-temperature superconductivity: Results from renormalized mean-field theory and variational Monte Carlo calculations

Abstract

Abstract 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. Acknowledgements The authors thank P. W. Anderson for several discussions and comments on this manuscript. VNM thanks P. W. Anderson, G. Baskaran, G. Levine, N. P. Ong, T. V. Ramakrishnan, D. Schmeltzer and Z.-Y. Weng for several discussions over the years. Notes The copper ion is in a d 9 configuration, with a single hole in the d-shell per unit cell. As shown by Zhang and Rice Citation25 this situation corresponds to a half-filled band in an effective single-band model. The superconducting phase is often divided into an optimal doped (doping level with highest T c), an overdoped (doping level higher than optimal doped) and an underdoped (doping level lower than optimal doped) regime. The weak coupling BCS ratio for s-wave superconductors, 2Δ / (k B T c)≈3.5. When speaking about (the magnitude of) the superconducting gap Δ in a d-wave state without specifying the momentum k, we mean the size of the gap at Henceforth we refer to the one-band Hubbard model by the phrase 'Hubbard model' For more details we refer the interested reader to the review articles by Moriya and Ueda Citation58, Yanase et al. Citation59 and Chubukov et al. Citation60. Long before the discovery of HTSCs Anderson and Fazekas Citation13, Citation14 proposed the RVB liquid as a possible ground state for the Heisenberg model on a 2D triangular lattice. The single hole problem together with the corresponding literature is discussed in Citation19 in more detail. A possible ansatz for finite temperatures was recently proposed by Anderson Citation113. He suggests a spin-charge locking mechanism within the Gutzwiller–RVB theory to describe the pseudogap phase in the underdoped cuprates as a vortex liquid state. For a real space representation of equation (Equation11) we refer to section 5.1.1. To avoid confusion, in this section we denote density operators with a 'hat' and write, e.g., Strictly speaking, we violate this rule by neglecting decouplings which include on-site pairing, . However, we work in the fully projected limit, i.e. |Ψ⟩ does not allow for on-site pairing. It is thus reasonable to prohibit on-site pairing in as well and to set The BCS states are defined by and , where and (s-wave, ; d-wave, ) We note that the inclusion of t" = − t' / 2 into the bare dispersion is sometimes used to get a better fit with band-structure calculations and ARPES data. Nevertheless VMC calculations Citation171 for a non-magnetic impurity report some minor discrepancies to the corresponding RMFT study Citation110. The Pfaffian is the analogue of a determinant which is defined only for antisymmetric matrices. For an antisymmetric matrix A, the square of the Pfaffian is equivalent to its determinant, namely, P f (A)² = |A|. Owing to the fixed particle number, the chemical potential μ becomes an additional free parameter. However, this parameter was fixed in Citation17 by setting the chemical potential μ to those of the unprojected wave function. For a more detailed reasoning leading to this step, we refer the reader to Citation206. Marshall Citation241 showed that the ground state of the spin- Heisenberg Hamiltonian on any bipartite lattice will be a singlet. Furthermore, the ground state wave function picks up a sign whenever two antiparallel spins from different sublattices are interchanged. This is the Marshall sign rule.