Zero-temperature resistive transition in Josephson-junction arrays at irrational
frustration
Enzo Granato
Laboratório Associado de Sensores e Materiais,
Instituto Nacional de Pesquisas Espaciais,
12245-970 São José dos Campos, SP Brazil
We use a driven Monte Carlo dynamics in the phase representation to determine the linear re-
sistivity and current-voltage scaling of a two-dimensional Josephson-junction array at an irrational
flux quantum per plaquette. The results are consistent with a phase-coherence transition scenario
where the critical temperature vanishes. The linear resistivity is nonzero at any finite tempera-
tures but nonlinear behavior sets in at a temperature-dependent crossover current determined by
the thermal critical exponent. From a dynamic scaling analysis we determine this critical exponent
and the thermally activated behavior of the linear resistivity. The results are in agreement with
earlier calculations using the resistively shunted-junction model for the dynamics of the array. The
linear resistivity behavior is consistent with some experimental results on arrays of superconducting
grains but not on wire networks, which we argue have been obtained in a current regime above the
crossover current.
PACS numbers: 74.81.Fa, 74.25.Qt, 75.10.Nr
Most theoretical investigations of the vortex-glass
phase in superconductors have considered model systems
where there is a combined effect of quenched disorder and
frustration1. However, in artificial Josephson-junction
arrays, frustration without disorder can in principle be
introduced by applying an external magnetic field on
a perfect periodic array of weakly coupled supercon-
ducting grains2,3,4 and similarly on superconducting wire
networks5,6. The frustration parameter f , the number of
flux quantum per plaquette, is given by f = φ/φo, the
ratio of the magnetic flux through a plaquette φ to the su-
perconducting flux quantum φo = hc/2e. It can be tuned
by varying the strength of the external field. Frustration
effects can be viewed as resulting from a competition be-
tween the underlying periodic pinning potential of the
array and the natural periodicity of the vortex lattice7.
At a rational value of f , the ground state is a commensu-
rate pinned vortex lattice leading to discrete symmetries
in addition to the continuous U(1) symmetry of the su-
perconducting order parameter. The resistive transition
is only reasonably well understood for simple rational
values of f .
At irrational values of f , the resistive behavior is much
less understood since the vortex lattice is now incom-
mensurate with the periodic array. In early Monte Carlo
(MC) simulations8 the ground state was found to con-
sist of a disordered vortex pattern lacking long range or-
der which could be regarded as a some sort of vortex-
glass state without quenched disorder. Glassy-like be-
havior was indeed observed in these simulations suggest-
ing a possible superconducting (vortex-glass) transition
at finite temperatures. However, some arguments also
suggested that the critical temperature should vanish7,9.
Simulations of the current-voltage scaling using the resis-
tively shunted-junction model for the dynamics of the ar-
ray found that the behavior was consistent with an equi-
librium resistive transition where the critical temperature
vanishes10, similar to the resistive transition described by
the the gauge-glass model in two dimensions1,11, but with
different values for the correlation-length critical expo-
nent ν. The linear resistivity is nonzero at any finite tem-
peratures but nonlinear behavior sets in at a crossover
current with a temperature dependence determined by
the exponent ν. This zero-temperature transition leads
to slow relaxation dynamics where the correlation length
diverges as a power law and the relaxation time diverges
exponentially as the temperature vanishes.
Simulations of the relaxation dynamics12 found a be-
havior analogous to relaxation in supercooled liquids with
a characteristic dynamic crossover temperature rather
than an equilibrium transition temperature, which is not
inconsistent with the zero-temperature transition sce-
nario. On the other hand, a systematic study by MC
simulations13 of a sequence of rational values of f con-
verging to the irrational frustration, using the vortex rep-
resentation, found two phase transitions at finite temper-
atures, a vortex-order transition weakly dependent on
f and a vortex pinning transition at much lower tem-
peratures varying with f , which should correspond to
the resistive transition. These results are in qualitative
agreement with MC simulations using the phase repre-
sentation of the same model14 but different ground states
were found.
More recently, MC simulations for the the specific
heat and relaxation dynamics found an intrinsic finite-
size effect15. The corresponding scaling analysis sug-
gested a zero-temperature transition with a critical ex-
ponent ν consistent with the value obtained initially
from current-voltage scaling10. However, a study of
the low-temperature configurations for frustrations close
the irrational value by MC simulations in the vortex
representation16, find two phase transitions consistent
with earlier work13.
On the experimental side, some results on arrays of su-
http://arxiv.org/abs/0704.1097v1
perconducting grains at irrational frustration2,3 are con-
sistent with the scenario of the zero-temperature resis-
tive transition but on wire networks5,6, resistivity scaling
showed evidence of a transition at finite temperature. Re-
cently, resistivity scaling suggesting a finite temperature
transition was also observed in arrays of superconducting
grains4.
In view of these conflicting results, it seems useful to
further investigate the current-voltage scaling for the ar-
ray at irrational frustration by studying both the non-
linear and linear resistivity with an improved method17
taking into account the long relaxation times. In fact, as
found recently, current-voltage scaling turned out to be
quite reliable in determining the phase-coherence transi-
tion even for a model with quenched disorder, such as the
three-dimensional XY-spin glass model17,18. The main
question is therefore, if the array at irrational frustration
displays an equilibrium phase-coherence transition at a
nonzero critical temperature into a state with vanishing
linear resistivity or its critical temperature vanishes and
the linear resistivity is finite at nonzero temperatures.
In this work, we investigate the resistivity scaling of
Josephson-junction arrays at a irrational frustration f =
5)/2, a golden irrational, using a driven MC dy-
namics in the phase representation introduced recently17.
The results are consistent with a phase-coherence tran-
sition scenario where the critical temperature vanishes,
Tc = 0. The linear resistivity is finite at nonzero temper-
atures but nonlinear behavior sets in at a temperature-
dependent crossover current determined by the thermal
critical exponent ν. The results agree with earlier simula-
tions using the resistively shunted-junction model for the
dynamics of the array10. However, with the present MC
method we are able to reach much lower temperatures
and current densities, improving the analysis of resistiv-
ity scaling and the estimate of the critical exponent ν. We
also argue that the finite-temperature transition found
in resistivity measurements on wire networks5,6have been
obtained in a current regime above the crossover current.
We consider a two-dimensional Josephson-junction
square array described by the Hamiltonian
H = −Jo
cos(θi − θj −Aij)− J
(θi − θi+x) (1)
The first term gives the Josephson-coupling energy be-
tween nearest neighbor grains where line integral of the
vector potential Aij is constrained to
ij Aij = 2πf
around each plaquette. The second term represents the
effects of an external driving current density J applied
in the x direction. When J 6= 0, the total energy is un-
bounded and the system is out of equilibrium. The lower-
energy minima occur at phase differences θi−θi+x which
increases with time t, leading to a net phase slippage
rate proportional to < d(θi − θi+x)/dt >, corresponding
to the voltage Vi,i+x. We take the frustration parameter
f equals an irrational number, f = (3−
5)/2, related to
the Golden Ratio Φ = (1+
5)/2 as f = 1− 1/Φ. In the
numerical simulations we use periodic (fluctuating twist)
boundary conditions on lattices of linear sizes L and cor-
responding rational approximations Φ = Fn+1/Fn, where
Fn are Fibonacci numbers (13, 21, 34, 55), with L = Fn.
To study the current-voltage scaling, we use a driven
MC dynamics method17. The time dependence is ob-
tained by identifying the MC time as the real time t and
we set the unit of time dt = 1, corresponding to a com-
plete MC pass through the lattice. Periodic (fluctuat-
ing twist) boundary conditions are used19. This bound-
ary condition adds new dynamical variables, uα (α = x
and y), corresponding to a uniform phase twist between
nearest-neighbor sites along the principal axis directions
x̂ and ŷ. A MC step consists of an attempt to change the
local phase θi and the phase twists uα by fixed amounts,
using the Metropolis algorithm. If the change in en-
ergy is ∆H , the trial move is accepted with probability
min{1, exp(−∆H/kT )}. The external current density J
in Eq. 1 biases these changes, leading to a net voltage
(phase slippage rate) across the system given by
(θ1,j − θL+1,j − uxL), (2)
in arbitrary units. The main advantage of this MC
method compared with the Langevin dynamics used
earlier10 is that in principle much longer time scales can
be accessed which allows one to obtain reliable data at
much lower temperatures and current densities. We have
determined the electric field E = V/L and nonlinear re-
sistivity ρ = E/J as a function of the driving current
density J for different temperatures T and different sys-
tem sizes. We used typically 2×105 MC steps to reach the
nonequilibrium steady state at finite current and equal
time steps to perform time averages with and additional
average over 4− 6 independent runs.
We have also determined the linear resistivity, ρL =
limJ−>0E/J , from equilibrium MC simulations. As any
transport coefficient, this quantity can be obtained from
equilibrium fluctuations and therefore can be calculated
in absence of an imposing driving current (J = 0). From
Kubo formula, the linear resistivity (resistance in two
dimensions) is given in terms of the equilibrium voltage
autocorrelation as
dt〈V (t)V (0)〉. (3)
Since the total voltage V is related to the phase difference
across the system ∆θ(t) by V = d∆θ(t)/dt, we find more
convenient to determine ρL from the long-time equilib-
rium fluctuations11 of ∆θ(t) as
〈(∆θ(t) −∆θ(0))2〉, (4)
which is valid for sufficiently long times t. To insure that
only equilibrium fluctuations are considered, the calcu-
lations were performed in two steps. First, simulations
using the exchange MC method (parallel tempering)20
 T=0.3
 T=0.275
 T=0.25
 T=0.225
 T=0.20
 T=0.175
 T=0.15
FIG. 1: Nonlinear resistivity E/J at different temperatures
T for system size L = 55.
were used to obtain equilibrium configurations of the sys-
tems at different temperatures21. This method is known
to reduce significantly the critical slowing down near the
transition allowing fully equilibration in finite small sys-
tem sizes. These configurations were then used as ini-
tial states for the driven MC dynamics process described
above, with J = 0, in order to obtain the ρL. The ini-
tial states are similar to the low-temperature states ob-
tained previously13,16 including thermal excitations. In
the parallel-tempering method20, many replicas of the
system with different temperatures are simulated simul-
taneously and the corresponding configurations are al-
lowed to be exchanged with a probability satisfying de-
tailed balance. The equilibration time can be measured
as the average number of MC steps required for each
replica to travel over the whole temperature range. We
used typically 4× 106 (parallel tempering) MC steps for
equilibration which is much larger than the estimated
equilibration time in systems with up to 100 replicas.
Subsequent MC simulations for the linear resistivity ob-
tained from Eq. 4 were performed using 2 × 103 time
averages for 2× 105 MC steps which is much larger than
the equilibrium relaxation time.
Fig. 1a shows the nonlinear resistivity E/J as a func-
tion of temperature for the largest system size. At small
current densities J , the nonlinear resistivity E/J tends
to a constant value, corresponding to the linear resis-
tivity ρL, which decreases rapidly with decreasing tem-
perature. For increasing J , the resistivity cross over to
a nonlinear behavior at a characteristic current density
Jnl, which also decreases with decreasing temperature.
To verify that the nonzero values approached at low cur-
rents in Fig. 1 correspond indeed to the linear resistivity
ρL, we show in Fig. 2 the temperature dependence of
ρL obtained without current bias from Eq.(4) for dif-
ferent system sizes. ρL decreases with system size but
approaches nonzero values for the largest system size.
These values are in agreement with the corresponding
values at the lowest current in Fig. 1. Since the be-
3 4 5 6 7
 L=13
 L=21
 L=34
 L=55
FIG. 2: Temperature dependence of the linear resistivity for
different system sizes.
havior of the ρL for the largest system size on the log-
linear plot in Fig. 2 is a straight line, it indicates an
activated Arrhenius behavior, where the linear resistiv-
ity decreases exponentially with the inverse of tempera-
ture with a temperature-independent energy barrier, es-
timated as Eb ∼ 1.07. Such activated behavior suggests
that the linear resistivity can be very small at low tem-
peratures but nevertheless remains finite for all tempera-
tures T > 0 and therefore there is no resistive transition
at finite temperatures. However, as will be described
below, the system behaves as if a resistive transition
occurs at zero temperature, corresponding to a phase-
coherence transition where the critical temperature van-
ishes, Tc = 0.
The behavior of the linear resistivity can be related to
the equilibrium relaxation time for phase fluctuations.
Since the voltage is the rate of change of the phase,
a nonzero ρL requires measurements over a time scale
τ ∝ 1/ρL, corresponding to the relaxation time for phase
fluctuations. Thus, we expect that τ should also have
an activated behavior, increasing exponentially with the
inverse of temperature. To verify this behavior, we have
in addition calculated the relaxation time τ for different
temperatures from the autocorrelation function of phase
fluctuations C(t) as
C(0)2
dtC(t) (5)
using MC simulations with J = 0. The starting con-
figurations were taken from equilibrium configurations
obtained21 with the parallel tempering MC method20.
The results shown on the log-linear plot in Fig. 3 are in-
deed consistent with an activated behavior of τ with an
energy barrier Eb = 1.18 in reasonable agreement with
the value obtained for the linear resistivity in Fig. 2.
The behavior in Figs. 1, 2 and 3 has the main features
associated with a phase transition that only occurs at
zero temperature, Tc = 0, similar to the two-dimensional
gauge glass model of disordered superconductors1,11. In
this case the correlation length ξ is finite for T > 0 but it
increases with decreasing temperature as ξ ∝ T−ν , with
ν a critical exponent. The divergent correlation length
near the transition determines both the linear an nonlin-
ear resistivity behavior leading to current-voltage scaling
sufficiently close to the critical temperature and suffi-
ciently small driving current. To understand in detail
the behavior of the linear ρL and nonlinear resistivity ρ
we need a scaling theory for the resistive behavior. If the
data satisfy such scaling behavior for different driving
currents and temperatures, the critical temperature and
critical exponents of the underlying equilibrium transi-
tion at J = 0 can then be determined from the best data
collapse. A detailed scaling theory has been described
in the context of the current-voltage characteristics of
vortex-glass models1 but the arguments should also ap-
ply to the present case. The basic assumption is the
existence of a second order phase transition. Measurable
quantities should then scale with the diverging correla-
tion length ξ ∝ |T −Tc|−ν and relaxation time τ near the
critical point. The nonlinear resistivity E/J should then
satisfy the scaling form1
τ = g±(
), (6)
in two-dimensions, where g±(x) is a scaling function. The
+ and − signs correspond to T > Tc and T < Tc, respec-
tively. If Tc 6= 0, then to satisfy such scaling form, the
nonlinear resistivity curves on the log-log plot in Fig. 1
should have a positive curvature at small J , with E/J de-
creasing with decreasing J to a temperature dependent
value for T > Tc while for T < Tc, the curvature should
be negative, with E/J vanishing in the limit J → 0. The
data in Fig. 1 do not show a change in curvature even for
the lowest temperature, already suggesting the possibil-
ity of a resistive transition at much lower temperatures
or at Tc = 0. However, a full scaling analysis of the data
is required to show that a transition indeed occur with
Tc = 0. If Tc = 0, then the correlation length ξ ∝ T−ν
and the linear resistivity ρL are both finite at T > 0.
One can then consider the behavior of the dimensionless
ratio E/JρL which should satisfy the scaling form
T 1+ν
) (7)
where g is a scaling function with g(0) = 1. A crossover
from linear behavior, when g(x) ∼ 1, to nonlinear behav-
ior, when g(x) >> 1, occurs when x ∼ 1 which leads to a
characteristic current density at which nonlinear behav-
ior sets in decreasing with temperatures as a power law,
Jnl ∝ T/ξ ∝ T 1+ν . The scaling form in Eq. (7)contains a
single critical exponent ν and does not depend on the par-
ticular form assumed for the divergence of the relaxation
time τ . However, for sufficiently low temperatures, the
relaxation process is expected to be thermally activated1
with τ ∝ exp(Eb/kT ). This corresponds formally to a
dynamic exponent z → ∞, if power-law behavior is as-
sumed for the relaxation time τ ∝ ξz . From the scal-
ing form of Eq.(6), the linear resistivity should scale as
3 4 5 6 7
              τ �  e
              E
=1.18(2)
FIG. 3: Temperature dependence of the relaxation time τ of
phase fluctuations for system size L = 55.
 ν = 1.4
 T=0.30
 T=0.275
 T=0.25
 T=0.225
 T=0.20
 T=0.175
 T=0.15
J / T 
 1 + ν
FIG. 4: Scaling plot of the nonlinear resistivity in Fig. 1 for
ν = 1.4.
ρL ∝ 1/τ and therefore it is also expected to have an
activated behavior, τ ∝ exp(−Eb/kT ). In general, the
energy barrier Eb also scales with the correlation length
as Eb ∝ ξψ , which leads to a temperature-dependent bar-
rier Eb ∝ T−ψν . A pure Arrhenius behavior corresponds
to ψ = 0. The behavior of the nonlinear and linear resis-
tivity in Figs 1, 2 and the relaxation time in Fig. 3 are
quite consistent with these predictions from the scaling
theory of a zero-temperature transition.
If there is a zero-temperature transition, as suggested
by the behaviors in Figs. 1, 2 and 3, then the data for
the nonlinear resistivity should satisfy the scaling form
of Eq.(7), if finite-size effects are negligible, and the best
data collapse provides an estimate of the critical expo-
nent ν. We expect that finite-size effects are negligible
for the largest system size L = 55 in Fig. 1 since at
this length scale the behavior of the linear resistivity is
roughly independent of the size as can be seen from Fig.
2. Fig. 4 shows that indeed the data for the largest
system size satisfy this scaling form with ν ∼ 1.4± 0.2.
The nonlinear resistivity should also satisfy the ex-
pected finite-size behavior in smaller system sizes when
the correlation length ξ approaches the system size L.
According to finite-size scaling, the scaling function in
Eq. (7), should also depend on the dimensionless ratio
L/ξ and so to account for finite-size effects the nonlinear
resistivity should satisfy the scaling form
= ḡ(
T 1+ν
, L1/νT ). (8)
The scaling analysis of the whole nonlinear resistivity
data is rather complicated in this case since the scal-
ing function depends on two variables. To simplify
the analysis22 we first estimate the temperature and
finite-size behavior of the crossover current density Jnl
where nonlinear behavior sets in as the value of J where
E/JρL = C, a constant. Then, from Eq. (8), the finite-
size behavior of Jnl can be expressed in the scaling form
(1+ν)/ν = ¯̄g(L1/νT ). (9)
The best data collapse according to the scaling in Eq. (9)
provides an alternative estimate of the critical exponent
ν. Fig. 5 shows that indeed the values of Jnl for different
system sizes and temperatures satisfy this scaling form
with ν ∼ 1.4, in agreement with the estimate obtained
for the largest system in Fig. 4 size using Eq. (7).
In addition to the standard finite-size effects, which oc-
cur when the correlation length is comparable to the sys-
tem size, already taken into account in the scaling form
of Eq. (8), there are also intrinsic finite-size effects15
resulting from the rational approximations used for the
irrational value of f . Since we use rational approxima-
tions Φ = Fn+1/Fn, where Fn are Fibonacci numbers
(13, 21, 34, 55), with the system size set to L = Fn, this
amounts essentially to have different values of the frus-
tration, fL = 1 − 1/Φ, for different system sizes which
will only converge to the correct value f = (3 −
in the infinite-size limit. We have assumed that such ef-
fects are negligible in the above scaling analysis but they
should affect our estimate of the critical exponent ν. In
principle, this intrinsic effect could be taken into account
within the zero-temperature transition scenario by allow-
ing for a size-dependent critical temperature Tc(L) in the
scaling analysis15. Alternatively, we could regard it as a
crossover from the critical behavior at the true irrational
frustration (infinite-size limit) to a phase with an addi-
tional small frustration δf = fL − f which should act as
a relevant perturbation. In this case, the scaling func-
tion in Eq. (7) should also depend on the dimensionless
ratio ξ2δf and again a scaling analysis with more than
one variable is required. However, our present numeri-
cal data is not sufficiently accurate to separate this effect
from standard finite-size effects.
The present results for the linear and nonlinear resis-
tivity of the array at irrational frustration obtained by
the driven MC dynamics agree with earlier simulations of
the current-voltage scaling using the resistively shunted-
junction model for the dynamics of the array10, where a
zero-temperature resistive transition was suggested and
1 2 3 4 5 6
ν = 1.4
 L=13
 L=21
 L=34
 L=55J n
1 / ν
FIG. 5: Finite-size scaling plot of the crossover current den-
sity Jnl with ν = 1.4, for different system sizes L.
the critical exponent was estimated as ν = 0.9(2). Al-
though the later model is expected to be a more realistic
description for the dynamics of the array, the value of the
static critical exponent ν should be the same for both
models. In general, the dynamic exponent z may de-
pend on the particular dynamics but since the relaxation
time τ is found to diverge exponentially for decreasing
temperature it corresponds to z → ∞ for both dynam-
ics. The present estimate of ν = 1.4(2), however, should
be more reliable since it considers much lower tempera-
tures and current densities and larger system size. In-
terestingly, similar behavior for the resistive transition
has been found both numerically and experimentally for
two-dimensional disordered superconductors in a mag-
netic field described as a gauge-glass model1,11 but with
different value for critical exponent ν ∼ 2. It should
be noted however that the actual ground state at irra-
tional frustration (without disorder) can be quite differ-
ent, as the self similar structure which has already been
proposed5,23. As would be expected, the different nature
of ground state leads to the different values of the critical
exponent ν.
Although the above scaling analysis is consistent with a
zero temperature transition, on pure numerical grounds
the data in Figs 1 and 2 can not complete ruled out
a vortex-order or a phase-coherence transition at tem-
peratures much lower than T = 0.15. In fact, phase-
coherence transitions were found in MC simulations using
the Coulomb-gas presentation13 at temperatures as low
as T ∼ 0.03 for the sequence of rational approximations
fL of the irrational f but since they show considerable
variation with fL it is not clear if it will remain nonzero
in the large size limit. However, the lowest temperature
in Figs 1 and 2 is already much smaller than the apparent
freezing temperature Tf ∼ 0.25 observed in earlier MC
simulations8. Below Tf , a nonzero Edwards-Anderson
order parameter q(t) =< ~Si >
2, was observed, where
~S = (cos θ, sin θ) and the average was taken over the sim-
ulation times t. Although this could suggest a diverging
relaxation time τ ∝
q(t)dt near a finite temperature
Tc ∼ Tf , such long relaxation time can also result from a
zero-temperature transition (Tc = 0) as suggested by the
above scaling analysis since in this case τ diverges expo-
nentially with decreasing temperature, τ ∝ exp(Eb/kT ),
as shown in Fig. 3. For low enough temperatures, τ will
eventually be larger than any simulation or experimental
measuring time scale and an apparent (time dependent)
freezing transition could occur depending on the partic-
ular dynamics and system size.
Some experimental results on arrays of superconduct-
ing grains at irrational frustration2,3 are consistent with
the scenario of a zero-temperature resistive transition
since even at the lowest temperatures a zero-resistance
state was not observed in these experiments. On the
other hand, current-voltage scaling analysis of experi-
mental data on wire networks5,6 was found to be con-
sistent with a resistive transition at finite temperature.
We note, however, that although the equilibrium behav-
ior of wire networks can be described by the same model
of Eq. 1, the nonlinear dynamical behavior may be quite
different since the nodes of the network are connected
by continuous superconducting wires, instead of weak
links, leading to additional larger energy barriers for vor-
tex motion, not included in the model, and consequently
larger phase-coherence length ξ and relaxation time τ
when compared with weak links24. In this case, the
characteristic crossover current to the linear resistivity
regime at low temperatures due to thermal fluctuations,
Jnl ∝ kT/ξ, expected in the zero-temperature transition
scenario, may only occur at current scales too small to
be detected experimentally. Thus the resistive behavior
is observed in a current regime at higher currents where
it follows the mean-field theory result25 where a vortex-
glass transition is possible at finite temperatures. How-
ever, the zero-temperature resistive could in principle be
observed in specially prepared wire networks in the weak
coupling regime where the additional energy barrier for
vortex motion can be minimized26. Other effects, such
as weak disorder, which is inevitably present in both ex-
perimental systems, should also be considered. It could
provide a possible explanation for the finite-temperature
resistive transition observed recently in arrays of super-
conducting grains4.
In conclusion, we have investigated the resistivity scal-
ing of Josephson-junction arrays at a irrational frustra-
tion using a driven MC dynamics17. The results are con-
sistent with a phase-coherence transition scenario where
the critical temperature vanishes, Tc = 0. The linear
resistivity is finite at nonzero temperatures but nonlin-
ear behavior sets in at a crossover current determined
by the thermal critical exponent ν. The results agree
with earlier simulations using the resistively shunted-
junction model for the dynamics of the array10 and more
recent MC simulations taking into account the intrinsic
finite-size effect15. Although we have only studied the
array at a particular value of irrational frustration, the
golden mean, we believe that the conclusion of a zero-
temperature phase-coherence transition should be valid
for all irrationals but possibly with different values of
the thermal critical exponent ν. The main advantage of
studying the golden mean value is that it is considered
the farthest from the low-order rationals and so intrinsic
finite-effects should be smaller. However, other irrational
frustrations have also been studied numerically15,23 and
experimentally5. The resistive behavior probes mainly
the phase-coherence of the system and since we find that
phase coherence is only attained at zero temperature, we
can not address directly the question of the existence of
a vortex-order transition at finite temperatures. In fact,
vortex order does not require long-range phase coherence.
Therefore, a vortex-order transition at zero temperature
or at finite temperature is consistent with the present
work. However, in view of the results for the supercooled
relaxation12 suggesting an analogy to structural glasses
such transition may be expected at finite temperature
and in fact is consistent with MC simulations indicating
a first-order vortex transition13,14,16. Thus, the inter-
esting possibility arises where the array undergoes two
transitions for decreasing temperature, a finite-resistance
vortex-order transition at finite temperature and a su-
perconducting transition only at zero temperature. This
phase transition scenario and the predicted behavior of
the linear and nonlinear resistivity provides an interesting
experimental signature for a Josephson-junction array at
irrational frustration.
This work was supported by FAPESP (grant 03/00541-
0) and computer facilities from CENAPAD-SP.
1 R.A. Hyman, M. Wallin, M.P.A. Fisher, S.M. Girvin, and
A.P. Young, Phys. Rev. B 51, 15304 (1995); D.S. Fisher,
M.P.A. Fisher, and D.A. Huse, Phys. Rev. B 43 130 (1991).
2 J.P. Carini, Phys. Rev. B 38, 63 (1988).
3 H. S. J. Zant, H. A. Rijken, and J. E. Mooij, J. Low Temp.
Phys. 82, 67 (1991).
4 I.C Baek, Y.J.Y Yu, and M.Y Choi, Phys. Rev. B 69,
172501 (2004).
5 F.Yu, N.E. Israeloff, A.M. Goldman, and R. Bojko, Phys.
Rev. Lett. 68, 2535 (1992).
6 X.S. Ling, H.J. Lezec, M.J. Higgins, J.S. Tsai, J. Fujita, H.
Numata, Y. Nakamara, Y. Ochiai, C. Tang, P.M. Chaikin,
and S. Bhattacharya, Phys. Rev. Lett. 76, 2989 (1996).
7 S. Teitel and C. Jayaprakash, Phys. Rev. Lett. 51, 1999
(1983).
8 T. Halsey, Phys. Rev. Lett. 55, 1018 (1985).
9 M.Y. Choi and D. Stroud, Phys. Rev. B 35, 7109 (1987).
10 E. Granato, Phys. Rev. B 54, 9655 (1996).
11 E. Granato, Phys. Rev. B 58, 11161 (1998).
12 B. Kim and S.J. Lee, Phys. Rev. Lett. 78, 3709 (1997).
13 P. Gupta, S. Teitel, M.J.P. Gingras, Phys. Rev. Lett. 80,
105 (1998).
14 C. Denniston and C. Tang, Phys. Rev. B 60, 3163 (1999).
15 S.Y. Park, M.Y. Choi, B.J. Kim, G.S. Jeon, and J.S.
Chung, Phys. Rev. Lett. 85, 3484 (2000).
16 S.J. Lee, J.R. Lee, and B. Kim, Phys. Rev. Lett. 88, 025701
(2002).
17 E. Granato, Phys. Rev. B 69, 144203 (2004).
18 L.W. Lee and A.P. Young, Phys. Rev. Lett. 22, 227203
(2003).
19 W.M. Saslow, M. Gabay, and W.-M. Zhang, Phys. Rev.
Lett. 68, 3627 (1992).
20 K. Hukushima and K. Nemoto, J. Phys. Soc. Jpn. 65, 1604
(1996); E. Marinari and G. Parisi, Europhys. Lett. 19, 451
(1992).
21 E. Granato, unpublished.
22 C. Wengel and A.P. Young, Phys. Rev. B 56, 5918 (1997)
23 M.R. Kolahchi, Phys. Rev. B 59, 9569 (1999).
24 H. S. J. Zant, M.N. Webster, J. Romijn, and J.E. Mooij,
Phys. Rev. B 42, 2647 (1990).
25 G. Parisi, J. Phys. A 27, 7555 (1994).
26 M. Giroud, O. Buisson, Y.Y. Wang, and B. Pannetier, J.
Low Temp. Phys. 87, 683 (1992).

  We use a driven Monte Carlo dynamics in the phase representation to determine
the linear resistivity and current-voltage scaling of a two-dimensional
Josephson-junction array at an irrational flux quantum per plaquette. The
results are consistent with a phase-coherence transition scenario where the
critical temperature vanishes. The linear resistivity is nonzero at any finite
temperatures but nonlinear behavior sets in at a temperature-dependent
crossover current determined by the thermal critical exponent. From a dynamic
scaling analysis we determine this critical exponent and the thermally
activated behavior of the linear resistivity. The results are in agreement with
earlier calculations using the resistively shunted-junction model for the
dynamics of the array. The linear resistivity behavior is consistent with some
experimental results on arrays of superconducting grains but not on wire
networks, which we argue have been obtained in a current regime above the
crossover current.

Zero-temperature resistive transition in Josephson-junction arrays at irrational
frustration
Enzo Granato
Laboratório Associado de Sensores e Materiais,
Instituto Nacional de Pesquisas Espaciais,
12245-970 São José dos Campos, SP Brazil
We use a driven Monte Carlo dynamics in the phase representation to determine the linear re-
sistivity and current-voltage scaling of a two-dimensional Josephson-junction array at an irrational
flux quantum per plaquette. The results are consistent with a phase-coherence transition scenario
where the critical temperature vanishes. The linear resistivity is nonzero at any finite tempera-
tures but nonlinear behavior sets in at a temperature-dependent crossover current determined by
the thermal critical exponent. From a dynamic scaling analysis we determine this critical exponent
and the thermally activated behavior of the linear resistivity. The results are in agreement with
earlier calculations using the resistively shunted-junction model for the dynamics of the array. The
linear resistivity behavior is consistent with some experimental results on arrays of superconducting
grains but not on wire networks, which we argue have been obtained in a current regime above the
crossover current.
PACS numbers: 74.81.Fa, 74.25.Qt, 75.10.Nr
Most theoretical investigations of the vortex-glass
phase in superconductors have considered model systems
where there is a combined effect of quenched disorder and
frustration1. However, in artificial Josephson-junction
arrays, frustration without disorder can in principle be
introduced by applying an external magnetic field on
a perfect periodic array of weakly coupled supercon-
ducting grains2,3,4 and similarly on superconducting wire
networks5,6. The frustration parameter f , the number of
flux quantum per plaquette, is given by f = φ/φo, the
ratio of the magnetic flux through a plaquette φ to the su-
perconducting flux quantum φo = hc/2e. It can be tuned
by varying the strength of the external field. Frustration
effects can be viewed as resulting from a competition be-
tween the underlying periodic pinning potential of the
array and the natural periodicity of the vortex lattice7.
At a rational value of f , the ground state is a commensu-
rate pinned vortex lattice leading to discrete symmetries
in addition to the continuous U(1) symmetry of the su-
perconducting order parameter. The resistive transition
is only reasonably well understood for simple rational
values of f .
At irrational values of f , the resistive behavior is much
less understood since the vortex lattice is now incom-
mensurate with the periodic array. In early Monte Carlo
(MC) simulations8 the ground state was found to con-
sist of a disordered vortex pattern lacking long range or-
der which could be regarded as a some sort of vortex-
glass state without quenched disorder. Glassy-like be-
havior was indeed observed in these simulations suggest-
ing a possible superconducting (vortex-glass) transition
at finite temperatures. However, some arguments also
suggested that the critical temperature should vanish7,9.
Simulations of the current-voltage scaling using the resis-
tively shunted-junction model for the dynamics of the ar-
ray found that the behavior was consistent with an equi-
librium resistive transition where the critical temperature
vanishes10, similar to the resistive transition described by
the the gauge-glass model in two dimensions1,11, but with
different values for the correlation-length critical expo-
nent ν. The linear resistivity is nonzero at any finite tem-
peratures but nonlinear behavior sets in at a crossover
current with a temperature dependence determined by
the exponent ν. This zero-temperature transition leads
to slow relaxation dynamics where the correlation length
diverges as a power law and the relaxation time diverges
exponentially as the temperature vanishes.
Simulations of the relaxation dynamics12 found a be-
havior analogous to relaxation in supercooled liquids with
a characteristic dynamic crossover temperature rather
than an equilibrium transition temperature, which is not
inconsistent with the zero-temperature transition sce-
nario. On the other hand, a systematic study by MC
simulations13 of a sequence of rational values of f con-
verging to the irrational frustration, using the vortex rep-
resentation, found two phase transitions at finite temper-
atures, a vortex-order transition weakly dependent on
f and a vortex pinning transition at much lower tem-
peratures varying with f , which should correspond to
the resistive transition. These results are in qualitative
agreement with MC simulations using the phase repre-
sentation of the same model14 but different ground states
were found.
More recently, MC simulations for the the specific
heat and relaxation dynamics found an intrinsic finite-
size effect15. The corresponding scaling analysis sug-
gested a zero-temperature transition with a critical ex-
ponent ν consistent with the value obtained initially
from current-voltage scaling10. However, a study of
the low-temperature configurations for frustrations close
the irrational value by MC simulations in the vortex
representation16, find two phase transitions consistent
with earlier work13.
On the experimental side, some results on arrays of su-
http://arxiv.org/abs/0704.1097v1
perconducting grains at irrational frustration2,3 are con-
sistent with the scenario of the zero-temperature resis-
tive transition but on wire networks5,6, resistivity scaling
showed evidence of a transition at finite temperature. Re-
cently, resistivity scaling suggesting a finite temperature
transition was also observed in arrays of superconducting
grains4.
In view of these conflicting results, it seems useful to
further investigate the current-voltage scaling for the ar-
ray at irrational frustration by studying both the non-
linear and linear resistivity with an improved method17
taking into account the long relaxation times. In fact, as
found recently, current-voltage scaling turned out to be
quite reliable in determining the phase-coherence transi-
tion even for a model with quenched disorder, such as the
three-dimensional XY-spin glass model17,18. The main
question is therefore, if the array at irrational frustration
displays an equilibrium phase-coherence transition at a
nonzero critical temperature into a state with vanishing
linear resistivity or its critical temperature vanishes and
the linear resistivity is finite at nonzero temperatures.
In this work, we investigate the resistivity scaling of
Josephson-junction arrays at a irrational frustration f =
5)/2, a golden irrational, using a driven MC dy-
namics in the phase representation introduced recently17.
The results are consistent with a phase-coherence tran-
sition scenario where the critical temperature vanishes,
Tc = 0. The linear resistivity is finite at nonzero temper-
atures but nonlinear behavior sets in at a temperature-
dependent crossover current determined by the thermal
critical exponent ν. The results agree with earlier simula-
tions using the resistively shunted-junction model for the
dynamics of the array10. However, with the present MC
method we are able to reach much lower temperatures
and current densities, improving the analysis of resistiv-
ity scaling and the estimate of the critical exponent ν. We
also argue that the finite-temperature transition found
in resistivity measurements on wire networks5,6have been
obtained in a current regime above the crossover current.
We consider a two-dimensional Josephson-junction
square array described by the Hamiltonian
H = −Jo
cos(θi − θj −Aij)− J
(θi − θi+x) (1)
The first term gives the Josephson-coupling energy be-
tween nearest neighbor grains where line integral of the
vector potential Aij is constrained to
ij Aij = 2πf
around each plaquette. The second term represents the
effects of an external driving current density J applied
in the x direction. When J 6= 0, the total energy is un-
bounded and the system is out of equilibrium. The lower-
energy minima occur at phase differences θi−θi+x which
increases with time t, leading to a net phase slippage
rate proportional to < d(θi − θi+x)/dt >, corresponding
to the voltage Vi,i+x. We take the frustration parameter
f equals an irrational number, f = (3−
5)/2, related to
the Golden Ratio Φ = (1+
5)/2 as f = 1− 1/Φ. In the
numerical simulations we use periodic (fluctuating twist)
boundary conditions on lattices of linear sizes L and cor-
responding rational approximations Φ = Fn+1/Fn, where
Fn are Fibonacci numbers (13, 21, 34, 55), with L = Fn.
To study the current-voltage scaling, we use a driven
MC dynamics method17. The time dependence is ob-
tained by identifying the MC time as the real time t and
we set the unit of time dt = 1, corresponding to a com-
plete MC pass through the lattice. Periodic (fluctuat-
ing twist) boundary conditions are used19. This bound-
ary condition adds new dynamical variables, uα (α = x
and y), corresponding to a uniform phase twist between
nearest-neighbor sites along the principal axis directions
x̂ and ŷ. A MC step consists of an attempt to change the
local phase θi and the phase twists uα by fixed amounts,
using the Metropolis algorithm. If the change in en-
ergy is ∆H , the trial move is accepted with probability
min{1, exp(−∆H/kT )}. The external current density J
in Eq. 1 biases these changes, leading to a net voltage
(phase slippage rate) across the system given by
(θ1,j − θL+1,j − uxL), (2)
in arbitrary units. The main advantage of this MC
method compared with the Langevin dynamics used
earlier10 is that in principle much longer time scales can
be accessed which allows one to obtain reliable data at
much lower temperatures and current densities. We have
determined the electric field E = V/L and nonlinear re-
sistivity ρ = E/J as a function of the driving current
density J for different temperatures T and different sys-
tem sizes. We used typically 2×105 MC steps to reach the
nonequilibrium steady state at finite current and equal
time steps to perform time averages with and additional
average over 4− 6 independent runs.
We have also determined the linear resistivity, ρL =
limJ−>0E/J , from equilibrium MC simulations. As any
transport coefficient, this quantity can be obtained from
equilibrium fluctuations and therefore can be calculated
in absence of an imposing driving current (J = 0). From
Kubo formula, the linear resistivity (resistance in two
dimensions) is given in terms of the equilibrium voltage
autocorrelation as
dt〈V (t)V (0)〉. (3)
Since the total voltage V is related to the phase difference
across the system ∆θ(t) by V = d∆θ(t)/dt, we find more
convenient to determine ρL from the long-time equilib-
rium fluctuations11 of ∆θ(t) as
〈(∆θ(t) −∆θ(0))2〉, (4)
which is valid for sufficiently long times t. To insure that
only equilibrium fluctuations are considered, the calcu-
lations were performed in two steps. First, simulations
using the exchange MC method (parallel tempering)20
 T=0.3
 T=0.275
 T=0.25
 T=0.225
 T=0.20
 T=0.175
 T=0.15
FIG. 1: Nonlinear resistivity E/J at different temperatures
T for system size L = 55.
were used to obtain equilibrium configurations of the sys-
tems at different temperatures21. This method is known
to reduce significantly the critical slowing down near the
transition allowing fully equilibration in finite small sys-
tem sizes. These configurations were then used as ini-
tial states for the driven MC dynamics process described
above, with J = 0, in order to obtain the ρL. The ini-
tial states are similar to the low-temperature states ob-
tained previously13,16 including thermal excitations. In
the parallel-tempering method20, many replicas of the
system with different temperatures are simulated simul-
taneously and the corresponding configurations are al-
lowed to be exchanged with a probability satisfying de-
tailed balance. The equilibration time can be measured
as the average number of MC steps required for each
replica to travel over the whole temperature range. We
used typically 4× 106 (parallel tempering) MC steps for
equilibration which is much larger than the estimated
equilibration time in systems with up to 100 replicas.
Subsequent MC simulations for the linear resistivity ob-
tained from Eq. 4 were performed using 2 × 103 time
averages for 2× 105 MC steps which is much larger than
the equilibrium relaxation time.
Fig. 1a shows the nonlinear resistivity E/J as a func-
tion of temperature for the largest system size. At small
current densities J , the nonlinear resistivity E/J tends
to a constant value, corresponding to the linear resis-
tivity ρL, which decreases rapidly with decreasing tem-
perature. For increasing J , the resistivity cross over to
a nonlinear behavior at a characteristic current density
Jnl, which also decreases with decreasing temperature.
To verify that the nonzero values approached at low cur-
rents in Fig. 1 correspond indeed to the linear resistivity
ρL, we show in Fig. 2 the temperature dependence of
ρL obtained without current bias from Eq.(4) for dif-
ferent system sizes. ρL decreases with system size but
approaches nonzero values for the largest system size.
These values are in agreement with the corresponding
values at the lowest current in Fig. 1. Since the be-
3 4 5 6 7
 L=13
 L=21
 L=34
 L=55
FIG. 2: Temperature dependence of the linear resistivity for
different system sizes.
havior of the ρL for the largest system size on the log-
linear plot in Fig. 2 is a straight line, it indicates an
activated Arrhenius behavior, where the linear resistiv-
ity decreases exponentially with the inverse of tempera-
ture with a temperature-independent energy barrier, es-
timated as Eb ∼ 1.07. Such activated behavior suggests
that the linear resistivity can be very small at low tem-
peratures but nevertheless remains finite for all tempera-
tures T > 0 and therefore there is no resistive transition
at finite temperatures. However, as will be described
below, the system behaves as if a resistive transition
occurs at zero temperature, corresponding to a phase-
coherence transition where the critical temperature van-
ishes, Tc = 0.
The behavior of the linear resistivity can be related to
the equilibrium relaxation time for phase fluctuations.
Since the voltage is the rate of change of the phase,
a nonzero ρL requires measurements over a time scale
τ ∝ 1/ρL, corresponding to the relaxation time for phase
fluctuations. Thus, we expect that τ should also have
an activated behavior, increasing exponentially with the
inverse of temperature. To verify this behavior, we have
in addition calculated the relaxation time τ for different
temperatures from the autocorrelation function of phase
fluctuations C(t) as
C(0)2
dtC(t) (5)
using MC simulations with J = 0. The starting con-
figurations were taken from equilibrium configurations
obtained21 with the parallel tempering MC method20.
The results shown on the log-linear plot in Fig. 3 are in-
deed consistent with an activated behavior of τ with an
energy barrier Eb = 1.18 in reasonable agreement with
the value obtained for the linear resistivity in Fig. 2.
The behavior in Figs. 1, 2 and 3 has the main features
associated with a phase transition that only occurs at
zero temperature, Tc = 0, similar to the two-dimensional
gauge glass model of disordered superconductors1,11. In
this case the correlation length ξ is finite for T > 0 but it
increases with decreasing temperature as ξ ∝ T−ν , with
ν a critical exponent. The divergent correlation length
near the transition determines both the linear an nonlin-
ear resistivity behavior leading to current-voltage scaling
sufficiently close to the critical temperature and suffi-
ciently small driving current. To understand in detail
the behavior of the linear ρL and nonlinear resistivity ρ
we need a scaling theory for the resistive behavior. If the
data satisfy such scaling behavior for different driving
currents and temperatures, the critical temperature and
critical exponents of the underlying equilibrium transi-
tion at J = 0 can then be determined from the best data
collapse. A detailed scaling theory has been described
in the context of the current-voltage characteristics of
vortex-glass models1 but the arguments should also ap-
ply to the present case. The basic assumption is the
existence of a second order phase transition. Measurable
quantities should then scale with the diverging correla-
tion length ξ ∝ |T −Tc|−ν and relaxation time τ near the
critical point. The nonlinear resistivity E/J should then
satisfy the scaling form1
τ = g±(
), (6)
in two-dimensions, where g±(x) is a scaling function. The
+ and − signs correspond to T > Tc and T < Tc, respec-
tively. If Tc 6= 0, then to satisfy such scaling form, the
nonlinear resistivity curves on the log-log plot in Fig. 1
should have a positive curvature at small J , with E/J de-
creasing with decreasing J to a temperature dependent
value for T > Tc while for T < Tc, the curvature should
be negative, with E/J vanishing in the limit J → 0. The
data in Fig. 1 do not show a change in curvature even for
the lowest temperature, already suggesting the possibil-
ity of a resistive transition at much lower temperatures
or at Tc = 0. However, a full scaling analysis of the data
is required to show that a transition indeed occur with
Tc = 0. If Tc = 0, then the correlation length ξ ∝ T−ν
and the linear resistivity ρL are both finite at T > 0.
One can then consider the behavior of the dimensionless
ratio E/JρL which should satisfy the scaling form
T 1+ν
) (7)
where g is a scaling function with g(0) = 1. A crossover
from linear behavior, when g(x) ∼ 1, to nonlinear behav-
ior, when g(x) >> 1, occurs when x ∼ 1 which leads to a
characteristic current density at which nonlinear behav-
ior sets in decreasing with temperatures as a power law,
Jnl ∝ T/ξ ∝ T 1+ν . The scaling form in Eq. (7)contains a
single critical exponent ν and does not depend on the par-
ticular form assumed for the divergence of the relaxation
time τ . However, for sufficiently low temperatures, the
relaxation process is expected to be thermally activated1
with τ ∝ exp(Eb/kT ). This corresponds formally to a
dynamic exponent z → ∞, if power-law behavior is as-
sumed for the relaxation time τ ∝ ξz . From the scal-
ing form of Eq.(6), the linear resistivity should scale as
3 4 5 6 7
              τ �  e
              E
=1.18(2)
FIG. 3: Temperature dependence of the relaxation time τ of
phase fluctuations for system size L = 55.
 ν = 1.4
 T=0.30
 T=0.275
 T=0.25
 T=0.225
 T=0.20
 T=0.175
 T=0.15
J / T 
 1 + ν
FIG. 4: Scaling plot of the nonlinear resistivity in Fig. 1 for
ν = 1.4.
ρL ∝ 1/τ and therefore it is also expected to have an
activated behavior, τ ∝ exp(−Eb/kT ). In general, the
energy barrier Eb also scales with the correlation length
as Eb ∝ ξψ , which leads to a temperature-dependent bar-
rier Eb ∝ T−ψν . A pure Arrhenius behavior corresponds
to ψ = 0. The behavior of the nonlinear and linear resis-
tivity in Figs 1, 2 and the relaxation time in Fig. 3 are
quite consistent with these predictions from the scaling
theory of a zero-temperature transition.
If there is a zero-temperature transition, as suggested
by the behaviors in Figs. 1, 2 and 3, then the data for
the nonlinear resistivity should satisfy the scaling form
of Eq.(7), if finite-size effects are negligible, and the best
data collapse provides an estimate of the critical expo-
nent ν. We expect that finite-size effects are negligible
for the largest system size L = 55 in Fig. 1 since at
this length scale the behavior of the linear resistivity is
roughly independent of the size as can be seen from Fig.
2. Fig. 4 shows that indeed the data for the largest
system size satisfy this scaling form with ν ∼ 1.4± 0.2.
The nonlinear resistivity should also satisfy the ex-
pected finite-size behavior in smaller system sizes when
the correlation length ξ approaches the system size L.
According to finite-size scaling, the scaling function in
Eq. (7), should also depend on the dimensionless ratio
L/ξ and so to account for finite-size effects the nonlinear
resistivity should satisfy the scaling form
= ḡ(
T 1+ν
, L1/νT ). (8)
The scaling analysis of the whole nonlinear resistivity
data is rather complicated in this case since the scal-
ing function depends on two variables. To simplify
the analysis22 we first estimate the temperature and
finite-size behavior of the crossover current density Jnl
where nonlinear behavior sets in as the value of J where
E/JρL = C, a constant. Then, from Eq. (8), the finite-
size behavior of Jnl can be expressed in the scaling form
(1+ν)/ν = ¯̄g(L1/νT ). (9)
The best data collapse according to the scaling in Eq. (9)
provides an alternative estimate of the critical exponent
ν. Fig. 5 shows that indeed the values of Jnl for different
system sizes and temperatures satisfy this scaling form
with ν ∼ 1.4, in agreement with the estimate obtained
for the largest system in Fig. 4 size using Eq. (7).
In addition to the standard finite-size effects, which oc-
cur when the correlation length is comparable to the sys-
tem size, already taken into account in the scaling form
of Eq. (8), there are also intrinsic finite-size effects15
resulting from the rational approximations used for the
irrational value of f . Since we use rational approxima-
tions Φ = Fn+1/Fn, where Fn are Fibonacci numbers
(13, 21, 34, 55), with the system size set to L = Fn, this
amounts essentially to have different values of the frus-
tration, fL = 1 − 1/Φ, for different system sizes which
will only converge to the correct value f = (3 −
in the infinite-size limit. We have assumed that such ef-
fects are negligible in the above scaling analysis but they
should affect our estimate of the critical exponent ν. In
principle, this intrinsic effect could be taken into account
within the zero-temperature transition scenario by allow-
ing for a size-dependent critical temperature Tc(L) in the
scaling analysis15. Alternatively, we could regard it as a
crossover from the critical behavior at the true irrational
frustration (infinite-size limit) to a phase with an addi-
tional small frustration δf = fL − f which should act as
a relevant perturbation. In this case, the scaling func-
tion in Eq. (7) should also depend on the dimensionless
ratio ξ2δf and again a scaling analysis with more than
one variable is required. However, our present numeri-
cal data is not sufficiently accurate to separate this effect
from standard finite-size effects.
The present results for the linear and nonlinear resis-
tivity of the array at irrational frustration obtained by
the driven MC dynamics agree with earlier simulations of
the current-voltage scaling using the resistively shunted-
junction model for the dynamics of the array10, where a
zero-temperature resistive transition was suggested and
1 2 3 4 5 6
ν = 1.4
 L=13
 L=21
 L=34
 L=55J n
1 / ν
FIG. 5: Finite-size scaling plot of the crossover current den-
sity Jnl with ν = 1.4, for different system sizes L.
the critical exponent was estimated as ν = 0.9(2). Al-
though the later model is expected to be a more realistic
description for the dynamics of the array, the value of the
static critical exponent ν should be the same for both
models. In general, the dynamic exponent z may de-
pend on the particular dynamics but since the relaxation
time τ is found to diverge exponentially for decreasing
temperature it corresponds to z → ∞ for both dynam-
ics. The present estimate of ν = 1.4(2), however, should
be more reliable since it considers much lower tempera-
tures and current densities and larger system size. In-
terestingly, similar behavior for the resistive transition
has been found both numerically and experimentally for
two-dimensional disordered superconductors in a mag-
netic field described as a gauge-glass model1,11 but with
different value for critical exponent ν ∼ 2. It should
be noted however that the actual ground state at irra-
tional frustration (without disorder) can be quite differ-
ent, as the self similar structure which has already been
proposed5,23. As would be expected, the different nature
of ground state leads to the different values of the critical
exponent ν.
Although the above scaling analysis is consistent with a
zero temperature transition, on pure numerical grounds
the data in Figs 1 and 2 can not complete ruled out
a vortex-order or a phase-coherence transition at tem-
peratures much lower than T = 0.15. In fact, phase-
coherence transitions were found in MC simulations using
the Coulomb-gas presentation13 at temperatures as low
as T ∼ 0.03 for the sequence of rational approximations
fL of the irrational f but since they show considerable
variation with fL it is not clear if it will remain nonzero
in the large size limit. However, the lowest temperature
in Figs 1 and 2 is already much smaller than the apparent
freezing temperature Tf ∼ 0.25 observed in earlier MC
simulations8. Below Tf , a nonzero Edwards-Anderson
order parameter q(t) =< ~Si >
2, was observed, where
~S = (cos θ, sin θ) and the average was taken over the sim-
ulation times t. Although this could suggest a diverging
relaxation time τ ∝
q(t)dt near a finite temperature
Tc ∼ Tf , such long relaxation time can also result from a
zero-temperature transition (Tc = 0) as suggested by the
above scaling analysis since in this case τ diverges expo-
nentially with decreasing temperature, τ ∝ exp(Eb/kT ),
as shown in Fig. 3. For low enough temperatures, τ will
eventually be larger than any simulation or experimental
measuring time scale and an apparent (time dependent)
freezing transition could occur depending on the partic-
ular dynamics and system size.
Some experimental results on arrays of superconduct-
ing grains at irrational frustration2,3 are consistent with
the scenario of a zero-temperature resistive transition
since even at the lowest temperatures a zero-resistance
state was not observed in these experiments. On the
other hand, current-voltage scaling analysis of experi-
mental data on wire networks5,6 was found to be con-
sistent with a resistive transition at finite temperature.
We note, however, that although the equilibrium behav-
ior of wire networks can be described by the same model
of Eq. 1, the nonlinear dynamical behavior may be quite
different since the nodes of the network are connected
by continuous superconducting wires, instead of weak
links, leading to additional larger energy barriers for vor-
tex motion, not included in the model, and consequently
larger phase-coherence length ξ and relaxation time τ
when compared with weak links24. In this case, the
characteristic crossover current to the linear resistivity
regime at low temperatures due to thermal fluctuations,
Jnl ∝ kT/ξ, expected in the zero-temperature transition
scenario, may only occur at current scales too small to
be detected experimentally. Thus the resistive behavior
is observed in a current regime at higher currents where
it follows the mean-field theory result25 where a vortex-
glass transition is possible at finite temperatures. How-
ever, the zero-temperature resistive could in principle be
observed in specially prepared wire networks in the weak
coupling regime where the additional energy barrier for
vortex motion can be minimized26. Other effects, such
as weak disorder, which is inevitably present in both ex-
perimental systems, should also be considered. It could
provide a possible explanation for the finite-temperature
resistive transition observed recently in arrays of super-
conducting grains4.
In conclusion, we have investigated the resistivity scal-
ing of Josephson-junction arrays at a irrational frustra-
tion using a driven MC dynamics17. The results are con-
sistent with a phase-coherence transition scenario where
the critical temperature vanishes, Tc = 0. The linear
resistivity is finite at nonzero temperatures but nonlin-
ear behavior sets in at a crossover current determined
by the thermal critical exponent ν. The results agree
with earlier simulations using the resistively shunted-
junction model for the dynamics of the array10 and more
recent MC simulations taking into account the intrinsic
finite-size effect15. Although we have only studied the
array at a particular value of irrational frustration, the
golden mean, we believe that the conclusion of a zero-
temperature phase-coherence transition should be valid
for all irrationals but possibly with different values of
the thermal critical exponent ν. The main advantage of
studying the golden mean value is that it is considered
the farthest from the low-order rationals and so intrinsic
finite-effects should be smaller. However, other irrational
frustrations have also been studied numerically15,23 and
experimentally5. The resistive behavior probes mainly
the phase-coherence of the system and since we find that
phase coherence is only attained at zero temperature, we
can not address directly the question of the existence of
a vortex-order transition at finite temperatures. In fact,
vortex order does not require long-range phase coherence.
Therefore, a vortex-order transition at zero temperature
or at finite temperature is consistent with the present
work. However, in view of the results for the supercooled
relaxation12 suggesting an analogy to structural glasses
such transition may be expected at finite temperature
and in fact is consistent with MC simulations indicating
a first-order vortex transition13,14,16. Thus, the inter-
esting possibility arises where the array undergoes two
transitions for decreasing temperature, a finite-resistance
vortex-order transition at finite temperature and a su-
perconducting transition only at zero temperature. This
phase transition scenario and the predicted behavior of
the linear and nonlinear resistivity provides an interesting
experimental signature for a Josephson-junction array at
irrational frustration.
This work was supported by FAPESP (grant 03/00541-
0) and computer facilities from CENAPAD-SP.
1 R.A. Hyman, M. Wallin, M.P.A. Fisher, S.M. Girvin, and
A.P. Young, Phys. Rev. B 51, 15304 (1995); D.S. Fisher,
M.P.A. Fisher, and D.A. Huse, Phys. Rev. B 43 130 (1991).
2 J.P. Carini, Phys. Rev. B 38, 63 (1988).
3 H. S. J. Zant, H. A. Rijken, and J. E. Mooij, J. Low Temp.
Phys. 82, 67 (1991).
4 I.C Baek, Y.J.Y Yu, and M.Y Choi, Phys. Rev. B 69,
172501 (2004).
5 F.Yu, N.E. Israeloff, A.M. Goldman, and R. Bojko, Phys.
Rev. Lett. 68, 2535 (1992).
6 X.S. Ling, H.J. Lezec, M.J. Higgins, J.S. Tsai, J. Fujita, H.
Numata, Y. Nakamara, Y. Ochiai, C. Tang, P.M. Chaikin,
and S. Bhattacharya, Phys. Rev. Lett. 76, 2989 (1996).
7 S. Teitel and C. Jayaprakash, Phys. Rev. Lett. 51, 1999
(1983).
8 T. Halsey, Phys. Rev. Lett. 55, 1018 (1985).
9 M.Y. Choi and D. Stroud, Phys. Rev. B 35, 7109 (1987).
10 E. Granato, Phys. Rev. B 54, 9655 (1996).
11 E. Granato, Phys. Rev. B 58, 11161 (1998).
12 B. Kim and S.J. Lee, Phys. Rev. Lett. 78, 3709 (1997).
13 P. Gupta, S. Teitel, M.J.P. Gingras, Phys. Rev. Lett. 80,
105 (1998).
14 C. Denniston and C. Tang, Phys. Rev. B 60, 3163 (1999).
15 S.Y. Park, M.Y. Choi, B.J. Kim, G.S. Jeon, and J.S.
Chung, Phys. Rev. Lett. 85, 3484 (2000).
16 S.J. Lee, J.R. Lee, and B. Kim, Phys. Rev. Lett. 88, 025701
(2002).
17 E. Granato, Phys. Rev. B 69, 144203 (2004).
18 L.W. Lee and A.P. Young, Phys. Rev. Lett. 22, 227203
(2003).
19 W.M. Saslow, M. Gabay, and W.-M. Zhang, Phys. Rev.
Lett. 68, 3627 (1992).
20 K. Hukushima and K. Nemoto, J. Phys. Soc. Jpn. 65, 1604
(1996); E. Marinari and G. Parisi, Europhys. Lett. 19, 451
(1992).
21 E. Granato, unpublished.
22 C. Wengel and A.P. Young, Phys. Rev. B 56, 5918 (1997)
23 M.R. Kolahchi, Phys. Rev. B 59, 9569 (1999).
24 H. S. J. Zant, M.N. Webster, J. Romijn, and J.E. Mooij,
Phys. Rev. B 42, 2647 (1990).
25 G. Parisi, J. Phys. A 27, 7555 (1994).
26 M. Giroud, O. Buisson, Y.Y. Wang, and B. Pannetier, J.
Low Temp. Phys. 87, 683 (1992).
