4 S. Khlebnikov / Nuclear Physics B 887 (2014) 1–18
A brane construction suitable for modeling a SC nanowire has been proposed in [11]. It is
based on a system of D3 and D5 branes in type IIB string theory.
3
The key aspects of it are as
follows:
(i) The setup contains a lar
ge number N of coincident D3 branes and a D5 probe intersecting
them over a line. This breaks all supersymmetries. The direction along the line, x ≡ x
1
, corre-
sponds to the direction of the wire, and the two directions transverse to all branes, x
8
+ ix
9
,
to the SC order parameter. (ii) N is identified with the number of occupied channels (transverse
wavefunctions) in the wire, N = N
ch
.
4
(iii) Supercurrent corresponds to the D5 winding around
the D3s as one moves along x. If the branes actually intersect, the low-energy spectrum of 3/5
strings contains N species of massless (1 +1)-dimensional fermions (both left and right movers).
At low values of P , however, the intersection is unstable, and the D5 moves a finite distance aw
ay
from the D3s; this corresponds to a fully gapped, supercurrent-only state.
Supercurrent-only solutions ha
ve been found in [11]. The Chern–Simons term in the D5 action
has the effect that the solution with winding number W carries NW units of the D5 worldvol-
ume charge. A phase slip corresponds to the D5 crossing the D3s, with W changing by one.
Conservation of charge then implies that N fundamental strings, stretching between the D5 and
the D3s, must be produced. This can be seen, on
the one hand, as a version of the Hanany–
Witten effect [20] in string theory (creation of branes and strings at intersections) and, on the
other, as a parallel to the requirement of quasiparticle production noted earlier. This parallel al-
lo
ws one to identify the worldvolume charge with the supercurrent momentum P
s
(which, for a
supercurrent-only state, is also the total momentum), as follows:
P
s
= NW, (6)
where by convention P
s
is in units of the Fermi momentum. Instead of W , we will often use the
winding number density
q =2πW/L, (7)
where an extra factor 2π is added for convenience.
In the leading lar
ge-N limit, supercurrent-only solutions exist for all q, no matter how
large. At q above a certain q
m
, however, phase slips become energetically favorable, and
the supercurrent-only state unstable. The instability has nothing to do with depairing. Indeed,
q
m
∼ 1/R, where R is the length scale of the D3 metric (the only length scale seen by classical
gravity). Meanwhile, the value q = q
dep
corresponding to depairing is of order of the gap Δ, i.e.,
of order R in units of the string tension. One sees that the ratio q
dep
/q
m
scales to infinity in the
large N limit.
5
Instability of the supercurrent-only state at q>q
m
means that for
P>P
m
= Nq
m
L/2π (8)
3
Our approach is different from other holographic descriptions of superconductivity that have been proposed in the
literature. It is distinct from the one in [12–14] in that it does not use a bulk U(1) gauge field. (States with nonzero
supercurrent in the model of [12,13] have been considered in [15–18].) And our approach is distinct from the brane
construction of [19] in that it preserves the worldvolume gauge symmetry. We use the corresponding conserved charge
to describe the linear momentum (quantized in units of the Fermi momentum k
F
).
4
N
ch
is proportional to the cross-sectional area of the wire and in practice is of order of a few thousand, for the
thinnest wires available. The large value reflects the 3d character of the electron density of states; this is in contrast to SC
properties, which vary only along x.
5
This may explain why no depairing is seen in the calculation of [11].