We show that the $t\ensuremath{-}J$ Hamiltonian is not in general reduced to ${H}_{t\ensuremath{-}J}=H(S\ensuremath{\rightarrow},f)$, where $S\ensuremath{\rightarrow}$ and
f stand for independent $([S\ensuremath{\rightarrow},f]=0)$ SU(2) (spin) generators and spinless fermionic (holon) fields, respectively. The proof is based upon an identification of the Hubbard operators with the generators of the su
(2∣1) superalgebra in the degenerate fundamental representation and ensuing SU
(2∣1) path-integral representation of the partition function ${Z}_{t\ensuremath{-}J}.$
Starting from H. Fröhlich's second-quantized Hamiltonian for a
d-dimensional electron gas in interaction with lattice phonons describing the quantum vibrations of a metal, we present a rigorous mathematical derivation of the superconducting state, following the principles laid out originally in 1957 by J. Bardeen, L. Cooper and J. Schrieffer. As in the series of papers written on the subject in the 90es, of which the present paper is a continuation, the representation of ions as a uniform charge background allows for a
(1+d)-dimensional fermionic quantum-field theoretic reformulation of the model at equilibrium. For simplicity, we restrict in this article to
d=2 dimensions and zero temperature, and disregard effects due to electromagnetic interactions. Under these assumptions, we prove transition from a Fermi liquid state to a superconducting state made up of Cooper pairs of electrons at an energy level
Γϕ∼ℏωDe−π/mλ equal to the mass gap, expressed in terms of the Debye frequency
ωD, electron mass
m and coupling constant
λ. The dynamical
U(1)-symmetry breaking produces at energies lower than the energy gap
Γϕ a Goldstone boson, a non-massive particle described by an effective
(2+1)-dimensional non-linear sigma-model, whose parameters and correlations are computed. The proof relies on a mixture of general concepts and tools (multi-scale cluster expansions, Ward identities), adapted to this quantum many-body problem with its extended infra-red singularity located on the Fermi circle, and a specific
1/N-expansion giving the leading diagrams at intermediate energies. Ladder diagrams are proved to provide the leading behavior in the infra-red limit, in agreement with mean-field theory predictions.
Coupled-cluster theory is the method of choice for weakly correlated systems, but in strongly correlated systems where the mean-field reference is qualitatively poor, it can break down badly. These failures can be ameliorated by using symmetry-broken mean-field references. For systems that spontaneously break spin symmetry, we might favor an unrestricted reference. In systems that instead break number symmetry, we turn to Bogoliubov coupled cluster theory. In this work, we apply Bogoliubov coupled-cluster to Hamiltonians with an underlying su(2) algebra to study the pairing Hamiltonian and the spin XXZ and J1-J2 models by exploiting the correspondence between fermionic-pair operators and spin-1/2 operators. For the pairing problem, the reference is the usual Bardeen--Cooper--Schrieffer (BCS) quasiparticle vacuum, while for spin systems we use the analogous spin-BCS reference, which breaks Sz symmetry and provides a flexible starting point for strongly correlated regimes. Correlation is incorporated through a hierarchy of coupled-cluster approximations. Beyond energies, we compute the response density matrices for the evaluation of spin--spin correlation functions for the XXZ and J1-J2 Heisenberg models, and the pairing parameter and particle-number variance for the pairing Hamiltonian. We also derive the corresponding relaxed density matrices, including the effects of orbital relaxation. Benchmark calculations for the half-filled pairing Hamiltonian and spin models demonstrate that Bogoliubov coupled cluster provides a systematically improvable wave-function-based approach for su(2) Hamiltonians, including geometrically frustrated spin systems requiring complex and noncollinear BCS references.
We use a renormalized mean-field theory to study the Gutzwiller projected Bardeen-Cooper-Schrieffer states of the extended Hubbard model in the large
U limit or the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J\text{\ensuremath{-}}{J}^{\ensuremath{'}}$ model on a two-dimensional checkerboard lattice. At small ${t}^{\ensuremath{'}}∕t$, the frustration due to the diagonal terms of ${t}^{\ensuremath{'}}$ and ${J}^{\ensuremath{'}}$ does not alter the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave pairing symmetry, and the negative (positive) ${t}^{\ensuremath{'}}∕t$ enhances (suppresses) the pairing order parameter. At large ${t}^{\ensuremath{'}}∕t$, the ground state has an $s\text{\ensuremath{-}}s$ wave symmetry. At the intermediate ${t}^{\ensuremath{'}}∕t$, the ground state is
d+id or
d+is wave with time reversal symmetry broken.