Ground state of uniformly frustrated Josephson-junction arrays at irrational frustration
Mohammad R. Kolahchi
Institute for Advanced Studies in Basic Sciences, Gava Zang, P.O. Box 45195-159, Zanjan, Iran
and Center for Theoretical Physics and Mathematics, P.O. Box 11365-8486, Tehran, Iran
Received 19 August 1998
A conjectured outline of the structure of vortex lattices of uniformly frustrated Josephson-junction arrays at
irrational frustration is presented. S0163-18299907513-X
The structure of the energy landscape of uniformly frus-
trated Josephson-junction arrays is not yet well understood.
Such systems are usually studied when the frustration param-
eter which denotes the number of flux quanta of the external
field through the unit cell of the array is rational; f = p / q . It
is believed that, at least at some large q values of f, the
system may exhibit glassylike behavior.
1,2
Such rational
numbers are the best representatives of the system at irratio-
nal frustration.
Uniformly frustrated XY models, embodied in arrays of
Josephson junctions in constant perpendicular magnetic field,
are described in the Landau gauge by
H =-J
ij
cos
i
-
j
-J
ij
cos
i
-
j
-2 mf .
1
The first term of this Hamiltonian includes the energies of
nearest-neighbor sites forming bonds that are directed along
the x axis of the square array. The second term is for bonds
directed along the y axis; m is an integer expressing the x
coordinate of these bonds in units of the lattice constant. The
Josephson coupling constant J sets the energy scale, and is
taken to be positive. The Hamiltonian is invariant if f is
changed to - f reversing the direction of magnetic field or
if an integer is added to f adding and integral number of flux
quanta.
For a rational f, the structure of the ground state of the
above Hamiltonian forms a superlattice of vortices having
typically a qxq unit cell. A clue to why this occurs is
suggested by Eq. 1. At m equal to integer multiples of q,
the frustrating phase 2 mf disappears, and the magnetic pe-
riod becomes commensurate with the period of the underly-
ing lattice. From isotropy of the system qxq periodicity fol-
lows. This is certainly not proof of the formation of the unit
cell of the vortex lattice, indeed few exceptions exist,
3
yet it
indicates how the periodicity of frustration is manifested in
the superlattice of vortices.
Recently, it was shown that for a class of local minimum
states of Hamiltonian 1, the two-dimensional problem of
finding the vortex structure, i.e., phases at sites such as a , b ,
is reduced to a problem in one dimension.
4
The idea amounts
to proving that the phase correlation between sites a , b and
a +1,b +1, which produces the Halsey staircase state, can be
generalized to the case where suitable phase correlations are
established between sites a , b and a +n , b +1. In this way
one can construct the two-dimensional lattice row by row,
each row having the same one-dimensional 1D arrange-
ment, only shifted by n plaquettes relative to the row below
Figs. 1a and 1c. It is shown that this results in the lattice
of vortices in a given row, to sit in the potential relief of the
row below, leading to a low-energy structure of the 2D array,
with a qxq unit cell. The outcome is a highly ordered set of
phases; due to certain symmetries, the number of indepen-
dent phases is q /2 ( q -1)/2 for q odd.
5
When f is irrational, one deals with vortex lattices incom-
mensurate with the underlying lattice—no unit cell exists. In
this case, the usual strategy for studying the spectrum of Eq.
1, and the corresponding vortex structures, is to make use
of best rational approximants of f. As our prime example, we
consider the case of f = , the golden ratio.
As mentioned above, due to the symmetries of Eq. 1,
f = =( 5 +1)/2, f = -1, and f =2 - are equivalent. The
sequence, r
i
, of best rational approximants of f = are then
effectively given by
1
2
,
1
3
,
2
5
,
3
8
,
5
13
,
8
21
,
13
34
,
21
55
, •••
F
N
F
N+2
, ••• , 2
where F
N
denotes the N th Fibonacci number, with F
5
=5
and r
5
=
5
13
.
The study of the above sequence of rationals amounts to
approximating the incommensurate vortex lattice, with the
best most commensurate lattices of increasingly larger pe-
riods. One then hopes to gain insight into the structure of the
vortex lattices at f = . Although the sequence of ground
state energies, thus found, approaches the ground state en-
ergy E ( ) due to the continuity of the spectrum,
6
it is not
possible to infer any property of the ultimately infinite vortex
lattice from the finite vortex lattices found at each rational
approximant. This is basically because of the periodic
boundary conditions which stabilize a particular vortex struc-
FIG. 1. Vortex lattices for f =
3
8
. In a, n =4; the lattice is
effectively 82, but it is not stable: the vortices move to fill the
empty columns. The resulting lattice is depicted in b; it has an
energy per site of -1.2764 J. In c, the Halsey staircase state is
shown, n =1, and the energy per site is -1.28146 J.
PHYSICAL REVIEW B 1 APRIL 1999-II VOLUME 59, NUMBER 14
PRB 59 0163-1829/99/5914/95694/$15.00 9569 ©1999 The American Physical Society