pdf, 9 MiB - Infoscience - EPFL
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28 CHAPTER 1. INTRODUCTION<br />
A(r) by the usual relation:<br />
χ ij = |χ ij | exp<br />
[<br />
i π φ 0<br />
∫ rj<br />
r i<br />
]<br />
A(r) · dr , (1.24)<br />
and it describes an electron moving in an external magnetic field given by the<br />
gauge field A(r). Consequently, the θ ij order parameter breaks the time-reversal<br />
symmetry of the original t−J model. The second term in H MF is the usual<br />
BCS pairing order parameter, it can be singlet or triplet pairing. The pairing<br />
parameter breaks the one dimensional U(1) symmetry. For the d-wave RVB<br />
phase, ∆ i,j is a nearest neighbor d-wave pairing with opposite signs on the vertical<br />
and horizontal bonds. Finally, the last term of the hamiltonian couples the<br />
spin operator to a classical external magnetic field. This latter term breaks<br />
the SU(2) symmetry. When this latter order parameter is present, the ground<br />
state of Hamiltonian (1.23) is no longer a singlet. Finally, µ plays the role of<br />
a chemical potential and allows to control the number of particles in the nonprojected<br />
wavefunction. We can expect that each of these instabilities could be<br />
stabilized in doped Mott-insulator, since the order parameter are obtained by<br />
natural decoupling of the exchange term of the t−J Hamiltonian.<br />
Although the t−J is formulated in a very simple form, the nature of the<br />
quantum correlations makes its physics very rich. One question of crucial interest<br />
is the interplay between superconductivity and antiferromagnetism close to the<br />
insulating phase in the t−J model. The ground state of this model on the square<br />
lattice is known to be antiferromagnetic at half-filling and one of the important<br />
questions is what happens upon doping. Although all the approaches to these<br />
strong coupling problems involve approximations, and it is sometimes difficult to<br />
distinguish the artefact due to approximations from the true features of the model,<br />
for the case of the square lattice, both the variational Monte Carlo method (VMC)<br />
[17, 18, 19] and mean-field theories [20] have found a d-wave superconducting<br />
phase in the the t−J model and a phase diagram which accounts for most of the<br />
experimental features of the high-T c cuprates [21, 22].<br />
In the limit of vanishing doping (half-filling), the d-wave RVB state can be<br />
viewed as an (insulating) resonating valence bond (RVB) or spin liquid state. In<br />
fact, such a state can also be written (after a simple gauge transformation) as<br />
a staggered flux state (SFP) [20, 23], i.e. can be mapped to a problem of free<br />
fermions hopping on a square lattice thread by a staggered magnetic field.<br />
Upon finite doping, although such a degeneracy breaks down, the SFP remains<br />
a competitive (non-superconducting) candidate with respect to the d-wave RVB<br />
superconductor [24]. In fact, it was proposed by P.A. Lee and collaborators<br />
[25, 26, 27] that such a state bears many of the unconventional properties of the<br />
pseudo-gap normal phase of the cuprate superconductors. This simple mapping<br />
connecting a free fermion problem on a square lattice under magnetic field [28]<br />
to a correlated wave-function (see later for details) also enabled to construct<br />
more exotic flux states (named commensurate flux states) where the fictitious