pdf, 9 MiB - Infoscience - EPFL

28 CHAPTER 1. INTRODUCTION
A(r) by the usual relation:
χ ij = |χ ij | exp
[
i π φ 0
∫ rj
r i
]
A(r) · dr , (1.24)
and it describes an electron moving in an external magnetic field given by the
gauge field A(r). Consequently, the θ ij order parameter breaks the time-reversal
symmetry of the original t−J model. The second term in H MF is the usual
BCS pairing order parameter, it can be singlet or triplet pairing. The pairing
parameter breaks the one dimensional U(1) symmetry. For the d-wave RVB
phase, ∆ i,j is a nearest neighbor d-wave pairing with opposite signs on the vertical
and horizontal bonds. Finally, the last term of the hamiltonian couples the
spin operator to a classical external magnetic field. This latter term breaks
the SU(2) symmetry. When this latter order parameter is present, the ground
state of Hamiltonian (1.23) is no longer a singlet. Finally, µ plays the role of
a chemical potential and allows to control the number of particles in the nonprojected
wavefunction. We can expect that each of these instabilities could be
stabilized in doped Mott-insulator, since the order parameter are obtained by
natural decoupling of the exchange term of the t−J Hamiltonian.
Although the t−J is formulated in a very simple form, the nature of the
quantum correlations makes its physics very rich. One question of crucial interest
is the interplay between superconductivity and antiferromagnetism close to the
insulating phase in the t−J model. The ground state of this model on the square
lattice is known to be antiferromagnetic at half-filling and one of the important
questions is what happens upon doping. Although all the approaches to these
strong coupling problems involve approximations, and it is sometimes difficult to
distinguish the artefact due to approximations from the true features of the model,
for the case of the square lattice, both the variational Monte Carlo method (VMC)
[17, 18, 19] and mean-field theories [20] have found a d-wave superconducting
phase in the the t−J model and a phase diagram which accounts for most of the
experimental features of the high-T c cuprates [21, 22].
In the limit of vanishing doping (half-filling), the d-wave RVB state can be
viewed as an (insulating) resonating valence bond (RVB) or spin liquid state. In
fact, such a state can also be written (after a simple gauge transformation) as
a staggered flux state (SFP) [20, 23], i.e. can be mapped to a problem of free
fermions hopping on a square lattice thread by a staggered magnetic field.
Upon finite doping, although such a degeneracy breaks down, the SFP remains
a competitive (non-superconducting) candidate with respect to the d-wave RVB
superconductor [24]. In fact, it was proposed by P.A. Lee and collaborators
[25, 26, 27] that such a state bears many of the unconventional properties of the
pseudo-gap normal phase of the cuprate superconductors. This simple mapping
connecting a free fermion problem on a square lattice under magnetic field [28]
to a correlated wave-function (see later for details) also enabled to construct
more exotic flux states (named commensurate flux states) where the fictitious