**Dirac-like quasiparticles in a magnetic field**

In the ground state of a superconductor (absolute temperature *T* = 0
K), all the electrons pair up to form Cooper pairs, which collectively
form the condensate. At finite *T*, however, minimization of the
free energy requires a small fraction of the electrons to be unpaired (this leads
to a finite entropy). The unpaired electrons are called quasiparticles.
The energy of a quasiparticle (qp) created in the state with wavevector **k**
equals *E*(**k**). At temperature *T*, the density of
quasiparticles is thermally activated and given by the expression

*n*** _{k}**(

where *k*_{B} is Boltzmann’s constant. In *s*-wave
supercondcutors, the *minimum* qp energy is *E*(**k**) = *D*(*T*), where *D*(*T*) is the superconducting gap (*D*(*T*) is *T* dependent, but
nominally independent of the direction of **k**). Hence, the qp
density is exponentially small at low *T*.

The *d*-wave symmetry of the superconducting gap in the cuprates
presents a novel situation. As we move around the 2*D* Fermi Surface
(FS) (blue circle in left figure), the superconducting gap *D*(**k**) changes sign 4 times.
Hence it vanishes at 4 nodal points **Q**. In the vicinity of
each node, the contours of *E*(**k**) are elliptical (shown as red
ellipses). In terms of the wavevector **q** = **k**-**Q**
measured relative to **Q**, *E*(**q**) has the form (the variation
along **z **is insignificant)

*E*(**q**) = [ (v_{f} *q*_{1})^{2}
+ (v_{D} *q*_{2})^{2} ]^{1/2},

where v_{f }and v_{D} are velocity parameters (with v_{f}/v_{D}
= 8-10). As indicated in the left figure, the principle axes *q*_{1}
and *q*_{2} are normal and parallel, respectively, to the FS.

Because *E*(**q**) vanishes at the node, the qp density decreases as
*T*^{2} (the number of states contained in the area of the ellipse
shaded in yellow), which is much slower than the exponential decrease in *s*-wave
superconductors. The relatively large qp population presents an
experimental opportunity to probe their properties at low *T*.

The expression for *E*(**q**) implies that, along a general
direction **q**, the qp energy increases linearly with the magnitude *q*.
The right figure shows the Dirac cone described by *E* as the direction of
**q** is varied (the slope of the cone is given by the velocity parameters v_{f}
and v_{D}). In a weak magnetic field, a quasiparticle (red ball)
will move around the cone, staying on the same energy contour (this is what *naive*
semiclassical theory would say). Thus, the qp’s in the cuprates present
an unusual situation for studying the behavior of excitations that obey a
Dirac-like dispersion, especially in an intense magnetic field.

Let us look at the nature of the quasiparticle state in more detail.
Although the qp may be regarded as simply an electron, it is actually a
superposition of an electron and a hole state. We represent a state
occupied by an electron of momentum **k** and spin up by c_{k}_{,up}^{+
}(the dagger symbol indicates electron creation). Similarly, a hole
state with the same momentum and spin is represented by d_{k}_{,up}^{+}
(the hole excitation is equivalent to destroying a spin-*down* electron in
the state –**k**; it is customary to write d_{k}_{,up}^{+}
= c_{-k,down} ). The quasiparticle state, represented by g_{k}_{,1}^{+}
is a quantum superposition of these two states with amplitudes _{k}** _{k}**, respectively, viz.

g_{k}_{,0}^{+}
= _{k}_{k}_{,up}^{+} + v** _{k}** d

A companion qp state g_{k}_{,1}^{+
}of the same momentum **k**, but with spin down, is defined with _{k}** _{k}** interchanged.

The quasiparticle g_{k}_{,0}^{+
}has definite momentum **k** and spin up. However, it has both
particle-like and hole-like characteristics. One or the other may
dominate in a particular experiment.