Competition between local and nonlocal dissipation effects in
  two-dimensional quantum Josephson junction arrays
Competition between lo
al and nonlo
al dissipation e�e
ts in two-dimensional
quantum Josephson jun
tion arrays
T. P. Polak
Consiglio Nazionale delle Ri
er
he - Istituto Nazionale per la Fisi
a della Materia,
Complesso Universitario Monte S. Angelo, 80126 Naples, Italy and
Istituto di Ciberneti
a �E.Caianiello� del CNR, Via Campi Flegrei 34, I-80078 Pozzuoli, Italy.
T. K. Kope¢
Institute for Low Temperatures and Stru
ture Resear
h,
Polish A
ademy of S
ien
es, POB 1410, 50-950 Wro
law 2, Poland
We dis
uss the lo
al and nonlo
al dissipation e�e
ts on the existen
e of the global phase 
oher-
en
e transitions in two dimensional Josephson-
oupled jun
tions. The quantum phase transitions
are also examined for various latti
e geometries: square, triangular and honey
omb. The T = 0
super
ondu
tor-insulator phase transition is analyzed as a fun
tion of several 
ontrol parameters
whi
h in
lude self-
apa
itan
e and jun
tion 
apa
itan
e and both lo
al and nonlo
al dissipation
e�e
ts. We found the 
riti
al value of the nonlo
al dissipation parameter α1 depends on a geometry
of the latti
e. The 
riti
al value of the normal state 
ondu
tan
e seems to be di�
ult to obtain
experimentally if we take into 
onsideration di�erent damping me
hanisms whi
h are presented in
real physi
al systems.
PACS numbers: 74.50.+r, 67.40.Db, 73.23.Hk
I. INTRODUCTION
Ma
ros
opi
 quantum e�e
ts in two-dimensional
Josephson jun
tion arrays (JJA's) have been ex-
tensively studied both theoreti
ally
1,2,3,4,5,6,7
experimentally
8,9,10
during the last years. The quan-
tum nature of the phase of a super
ondu
ting order
parameter is re�e
ted in phase transitions in JJA's. In
nondissipative JJA's the two main energy s
ales are set
by the Josephson 
oupling EJ between super
ondu
ting
islands and the ele
trostati
 energy EC arising from lo
al
deviations from 
harge neutrality. The ratio EC/EJ
determines the relevan
e of the quantum �u
tuations
and when it in
reases above a 
riti
al value, the phase
order is destroyed and the array turns into the insulator.
For large 
apa
itive 
oupling EC ≫ EJ the system 
an
be modeled by a renormalized 
lassi
al two-dimensional
(2D) XY model. In the opposite limit, the energy 
ost
for transferring 
harges between neighboring islands in
the array is so high that 
harges tend to be lo
alized.
While the nature of the 
lassi
al 2D XY model is
well understood, its quantum generalization still poses
unsettled issues.
Modern fabri
ation te
hniques allow one to make ar-
rays of ultrasmall super
ondu
ting islands separated by
insulators. In su
h systems the important fa
tor whi
h
has a profound impa
t on the ground state of the JJA's
is dissipation 
aused by Ohmi
 resistors shunting the
jun
tions
11,12,13
or quasiparti
le tunneling through the
jun
tions.
14,15,16
Despite several experiments with 2D
JJA's
17,18
and super
ondu
ting granular �lms
exis-
ten
e of the dissipation driven transition and 
riti
al
value of the normal state 
ondu
tan
e is at least ques-
tionable.
Phase diagrams in quantum JJA's with both me
ha-
nisms of dissipation Ohmi
 and quasiparti
le were stud-
ied theoreti
ally by Zaikin.
Cal
ulations done within
the framework of the instanton te
hnique reveal a zero
temperature phase diagram with two dissipative phase
transitions. The author 
laims there are regions on the
phase diagram where disordered phase and the 
lassi-

al Josephson e�e
t 
ould take pla
e. Cu

oli et al.
presented an analyti
al study based on the e�e
tive po-
tential approa
h. They proposed a model in whi
h two
di�erent relaxation times lead to the 
ondu
tan
e matrix
with resistive shunts to the ground and among islands.
Despite of these a

urate analyti
al studies the problem
of a theoreti
al explanation of phase diagrams in JJA's
in dissipative environment is still open.
Our previous theoreti
al work
in whi
h the attention
was fo
used on lo
al dissipation e�e
ts, predi
ted the ex-
isten
e of the 
riti
al value of the dissipation parameter
α = 2 independent on geometry of a latti
e and mag-
neti
 �eld. Other theoreti
al studies
11,12,23,24
suggest a
rather broad range of the 
riti
al values of the dissipation
parameter α = 0.5, 0.84, 1, 2 whi
h depends on the di-
mension of the system and me
hanism of the dissipation.
It seems that an unambiguous in experimental measure-
ment of the 
riti
al value of the normal state 
ondu
tan
e
is elusive. Several groups
17,18,25,26,27,28,29
using di�erent
experimental te
hniques obtained various 
riti
al values
of α = 0.5, 0.8, 1. To explain these theoreti
al and ex-
perimental di�
ulties we propose a model in whi
h lo
al
(
aused by shunt resistors 
onne
ting the islands to a
ground) and nonlo
al (shunt resistors in parallel to the
jun
tions) dissipation e�e
ts are 
onsidered.
The purpose of this paper is to investigate phase tran-
sitions at zero temperature in two-dimensional 
apa
i-
tively 
oupled super
ondu
ting arrays with emphasis on
http://arxiv.org/abs/0704.1993v1
the 
ompetition of lo
al α0 and nonlo
al α1 dissipation
e�e
ts. The detailed phase boundary 
ru
ially depends
on the ratio of mutual to self-
apa
itan
es C1/C0 and
spe
i�
 planar geometry of the array.
Aware of that fa
t
we 
onsider 
apa
itive matrix Cij within the range of
parameters C1/C0 whi
h 
an be adjusted to the exper-
imental samples. We analyze phase diagrams for three
di�erent latti
es: square (�), triangular (△) and honey-

omb (H). We want to emphasize that our approximation

annot be used for analysis of the Berezinski-Kosterlitz-
Thouless transitions sin
e it is appropriate only for phys-
i
al systems where long-range order appears.
The outline of the rest of the paper is the following: In
Se
. II we de�ne the model Hamiltonian, followed by its
path integral formulation in terms of the dimensionality
dependent nonmean-�eld like approa
h. In Se
 III we
present the zero-temperature phase diagram results for
di�erent JJA's geometries. Finally, in Se
. IV we dis
uss
our results and their relevan
e to other theoreti
al and
experimental works.
II. MODEL
We 
onsider a two-dimensional Josephson jun
tion ar-
ray with latti
e sites i, 
hara
terized by super
ondu
ting
phase φi in dissipative environment. The 
orresponding
Eu
lidean a
tion reads:
S = SC + SJ + SD, (1)
where
〈i,j〉
dτJij {1− cos [φi (τ) − φj (τ)]} ,
dτdτ ′αij (τ − τ ′) [φi (τ) − φj (τ ′)]
and τ is the Matsubara's imaginary time (0 ≤ τ ≤
1/kBT ≡ β); T is temperature and kB the Boltzmann

onstant (~ = 1). The �rst part of the a
tion (2) de-
�nes the ele
trostati
 energy where Cij is the 
apa
i-
tan
e matrix whi
h is a geometri
 property of the array.
This matrix is usually approximated as a diagonal (self-

apa
itan
e C0) and a mutual one C1 between nearest
neighbors. We 
an write a general expression for the Cij
in the following form:
Cij =
C0 + zC1 for i = j
−C1 for nearest neighbors
whi
h holds for periodi
 stru
tures in any dimension; z
is 
oordination number of the network. The se
ond term
is the Josephson energy EJ (Jij ≡ EJ for |i− j| = |d|
and zero otherwise). The ve
tor d forms a set of z latti
e
translation ve
tors, 
onne
ting a given site to its near-
est neighbors. The Fourier transformed wave-ve
tor de-
pendent Josephson 
ouplings Jk are di�erent for various
latti
es. The third part of the a
tion SD des
ribes the
dissipation e�e
ts and αij (τ − τ ′) is a dissipation ma-
trix. We 
hoose two independent damping me
hanisms,
the on-site and the nearest-neighbor, be
ause usually, the
damping is des
ribed in terms of shunt resistors R0 
on-
ne
ting the islands to a ground and shunt resistors in
parallel to the jun
tions related to R1. We 
an write
dissipation matrix similar to Eq. (3) in a more 
losed
form:
αij = (α0 + zα1) δij − α1
δi,j+d (4)
with the ve
tor d running over nearest neighboring is-
lands. The dimensionless parameters
, α1 =
. (5)
des
ribe strength of the lo
al and nonlo
al dissipation
respe
tively, where RQ = 1/4e
is quantum resistan
e.
A. Method
Most of existing analyti
al works on quantum JJA's
have employed di�erent kinds of mean-�eld-like approxi-
mations whi
h are not reliable for treatment spatial and
temporal quantum phase �u
tuations. The model in Eq.
2 en
odes the phase �u
tuation algebra given by Eu-

lidean group E2 de�ned by 
ommutation relations be-
tween parti
le Li and phase Pj operators,
Pj = e
iφj ,
[Li, Pj ] = −Piδij ,
Li, P
i δij ,
[Pi, Pj ] = 0, (6)
with the 
onserved quantity (invariant of the E2 algebra)
i ≡ P
xi + P
yi = 1. (7)
The proper theoreti
al treatment of the quantum JJA's
must maintain the 
onstraint in Eq. 7. A formulation
of the problem in terms of the spheri
al model initiated
by Kope¢ and José
leads us to introdu
e the auxiliary

omplex �eld ψi whi
h repla
es the original operator Pi.
Furthermore, relaxing the original �rigid� 
onstraint and
imposing the weaker spheri
al 
ondition:
i = N. (8)
where N is the number of latti
e sites, allows us to im-
plementation the spheri
al 
onstraint:
[Dψ] δ
|ψi|2 −N
e−SJ[ψ]
[Dφ] e−SC+D[φ]
δ [Reψi − Pxi (φ)]
×δ [Imψi − Pyi (φ)] . (9)
where [Dψ] =
iDψiDψ∗j and [Dφ] =
iDφi. It is 
on-
venient to employ the fun
tional Fourier representation
of the δ fun
tional to enfor
e the spheri
al 
onstraint in
Eq. (8):
δ [x (τ)] =
∫ +i∞
dτλ(τ)x(τ), (10)
whi
h introdu
es the Lagrange multiplier λ (τ) thus
adding a quadrati
 term (in ψ �eld) to the a
tion in Eq.
(2). The evaluation of the e�e
tive a
tion in terms of the
ψ to se
ond order in ψi gives the partition fun
tion of
the quantum spheri
al model (QSM)
ZQSM =
[Dψ] δ
|ψi|2 −N
e−S[ψ] (11)
where the e�e
tive a
tion reads:
S[ψ] =
〈i,j〉
dτdτ ′ {[Jij (τ) δ (τ − τ ′)
+ W−1ij (τ, τ
′)− λ (τ) δijδ (τ − τ ′)
+ Nλ (τ) δ (τ − τ ′)} . (12)
Furthermore,
Wij (τ, τ ′) =
[Dφ] ei[φi(τ)−φj(τ
′)]e−SC+D[φ], (13)
is the phase-phase 
orrelation fun
tion with statisti
al
[Dφ] e−SC+D[φ], (14)
where a
tion SC+D [φ] is just a sum of ele
trostati
 and
dissipative terms in Eq. (2). After introdu
ing the
Fourier transform of the �eld
φi (τ) =
φk,ne
−i(ωnτ−kri)
with ωn = 2πn/β, (n = 0,±1,±2, ...) being the Bose
Matsubara frequen
ies. From Eq. (13) the phase-phase

orrelation fun
tion reads:
W (τ, τ ′) = exp
1− cos [ωn (τ − τ ′)]
ω2n +
. (16)
The 
harging energy parameter entering Eq. (16) is
π (C0 + 4C1)
C0 + 4C1
where K (x) is the ellipti
 integral of the �rst kind32.
Furthermore we introdu
e quantities E0 = e
2/2C0 and
E1 = e
2/2C1 related to the island and jun
tion 
apa
i-
tan
es.
The dissipative parameter α and may be expli
itly
written as
α−1 = lim
α0 + zα1 − 2α1Ek
where Ek is a dispersion and has di�erent form for various
latti
es. In the present paper we 
onsider three di�erent
geometries of the latti
e: square (�), triangular (△) and
honey
omb (H):
= cos kx + cos ky,
= cos kx + 2 cos
. (19)
with the latti
e spa
ing set to 1. The results of the sum
over wave ve
tors in Eq. 18 are pla
ed in Appendix B.
Finally, for small frequen
ies, α0 ≤ 2 and α1 ≤ 1 the
inverse of 
orrelation fun
tion (16) be
omes:
W−1 (ωn) =
ω2n +
|ωn| for ωn 6= 0
0 otherwise
In order to determine the Lagrange multiplayer λ we ob-
serve that in the thermodynami
 limit (N → ∞) the
steepest des
ent method be
omes exa
t. The 
ondition
the integrand in Eq. (11) has a saddle point λ (τ) = λ0,
leads to an impli
it equation for λ0:
G (k, ωn) , (21)
where
G−1 (k, ωn) = λ0 − Jk +
ω2n +
|ωn| . (22)
The emergen
e of the 
riti
al point in the model is sig-
naled by the 
ondition
G−1 (k = 0, ωn = 0) ≡ λ0 − J0 = 0 (23)
whi
h �xes the saddle point of the Lagrange multiplier
λ0 within the ordered phase λ0 = J0.
0.3 0.6 0.9 1.2 1.5 1.8 2.1
Figure 1: Zero-temperature phase diagram for the total 
harg-
ing energy EJ/EC vs parameter of dissipation αi (i = 1 if
α0 = 0 and i = 0 if α1 = 0) for triangular (△; α
1 = 2/3),
square (�; αcrit1 = 1) and honey
omb (H ; α
1 = 4/3) lat-
ti
e. Insulating (super
ondu
ting) state is below (above) the

urves.
III. PHASE DIAGRAMS
A Fourier transform of the Green fun
tion in Eq. (22)
enables one to write the spheri
al 
onstraint (21) expli
-
itly as:
ρ (ξ)
λ− ξEJ + 18EC ω
ξ |ωn|
. (24)
where
ρ (ξ) =
is the density of states. We 
an easily see that solu-
tion of the model requires the knowledge of the DOS
for a spe
i�
 latti
e with the superimposition of the self-

onsisten
y 
ondition for the 
riti
al line in Eq. (24).
A Josephson-jun
tion array network is 
hara
terized for
di�erent latti
es by the nearest-neighbor Josephson 
ou-
pling EJ with the following wave-ve
tor dependen
e
= EJE�k
= EJE△k
= EJEHk (26)
where Ek's are given by Eq. 19. The Fourier transform
of the 
apa
itan
e (dissipative) matri
es for a triangular
and honey
omb latti
e

an be also found in Appendix
0.2 0.4 0.6 0.8 1 1.2
a a0 1 0= =
a0 2,≃ a1 1≃
PCSdiss
PCSdiss
E EJ0 /
E EJ1 /
crit crit
Figure 2: Zero-temperature phase diagram for square 2D
JJA's with self C0 = e
2/2E0 and mutual C1 = e
2/2E1 
a-
pa
itan
e (Eq. 17) for two values of lo
al and nonlo
al dis-
sipation parameter α0 = α1 = 0 and α0 = 2 and α1 = 1
(see Appendix). We 
an distinguish three areas: phase 
oher-
ent state (PCS) where phases in the islands are well de�ned.
Insulating state (IS) whi
h 
ould be driven to the phase 
o-
herent state by e�e
ts of the dissipation (PCSdiss). Finally,
insulating state, where super
ondu
ting phase is perturbed by
strong zero point quantum �u
tuations due to Coulomb blo
k-
ade that lo
alizes 
harge 
arries to the islands. However sys-
tem 
an be driven to the phase 
oherent state (PCScritdiss) but
only by 
riti
al values of the dissipation parameters (αcrit0 ≃ 2
and αcrit1 ≃ 1).
By substituting the value of λ0 = Jmax where Jmax
denotes maximum value of the spe
trum Jk, and after
performing the summation over Matsubara frequen
ies,
in T → 0 limit we obtain the following result:
ρ (ξ)
)2 − Jmax−ξEJ
)2 − Jmax−ξEJ
)2 − Jmax−ξEJ
 . (27)
The 
riti
al values of the nonlo
al dissipation param-
eters have a sour
e in low temperature properties of
the JJA's 
orrelation fun
tion in dissipative environment
(Appendix A). The dependen
e of the 
riti
al value α1
depi
ted in Fig. 1 is a dire
t result of the divergen
e
this phase-phase 
orrelator. The Fig. 2 and Fig. 3 point
out the big di�eren
e in values of the self C0 and mutual
C1 
apa
itan
e and 
ompetition between various dissi-
pation me
hanisms have a severe impa
t on phase dia-
grams. In typi
al real situations mutual 
apa
itan
e 
an
be at least two orders of magnitude larger than the self-

apa
itan
e what indi
ates the samples are pla
ed very

lose to E1/EJ axis in Fig. 2.
JJA's devoid of dissipation e�e
ts 
an be in two phases:
insulator phase (IS) and phase 
oherent state (PCS).
However 
oupling system to the environment we are able
0.2 0.4 0.6 0.8 1 1.2
a0 PCSdiss
Figure 3: Zero-temperature phase diagram for square 2D
JJA's in spa
e of lo
al α0 and nonlo
al α1 dissipation pa-
rameters for several values of the ratio C1/C0. From the top
C1/C0 = 1.25, 1.67, 2.5, 5, 10, 50. Insulating (super
ondu
t-
ing) state below (above) the 
urves.
to drive arrays into PCS even if lo
alization of the 
harge

arriers due to Coulomb blo
kade is strong and dominates
properties of the system. Furthermore for ea
h geometry
of the latti
e there are 
riti
al values of the dissipation
parameters that lead arrays to situation (Fig. 2) where
phases in the islands are well de�ned and quantum �u
-
tuations do not perturb a super
ondu
ting phase - region
des
ribe as PCScritdiss. Notwithstanding between these two
boundary situations there is a region on the phase dia-
gram in Fig. 2 where 
on
rete situation depends on the
values of the parameters. In this area PCSdiss system

an be driven to the PCS but 
oupling to the environ-
ment does not have to be so strong as in PCScritdiss 
ase.
IV. RESULTS
Until now only three papers 
onsidered e�e
ts with
both me
hanisms of the dissipation.
11,20,21
The most in-
teresting is Cu

oli's work where authors introdu
ed the
full 
ondu
tan
e matrix for triangular and square lat-
ti
es. It seems their results improve the quantitative a
-

ura
y; nevertheless problem of the theoreti
al explana-
tion of the phase diagram of JJA's in dissipative environ-
ments is thus open.
A model of an ordered array of resistively shunted
Josephson jun
tions was also 
onsidered by Chakravarty
et. al.
and simpli�ed at several points. They assumed
that 
apa
itan
e matrix is diagonal Cij = Cδij . The au-
thors 
laim the results do not depend sensitively on de-
tailed form of Cij . Moreover the matrix αij = h/4e
where Rij is the shunting resistan
e between grains i and
j is redu
ed to the form in whi
h the information about
the geometry of the latti
e is not in
luded. The obtained
zero-temperature phase diagram reveals the fa
t that the

riti
al value of the dissipation exists and is proportional
to the inverse of the dimension of the system whi
h gives
us 
riti
al value α = 1/2 for a square latti
e, but espe-

ially at low temperatures, variational methods are not
pre
ise enough to per
eive su
h a subtle transition.
In our model the 
riti
al value of the nonlo
al dissipa-
tion parameter α1 behaves similarly. It depends on the
maximum value of the Jk spe
trum. Be
ause Jk exhibits
di�erent 
hara
ters for various latti
es hen
e we 
ould
observe phenomenon su
h as nonmonotoni
 dependen
e
of the 
riti
al value of the nonlo
al dissipation parameter
for various geometries of the array (see Fig. 1). When
we assume diagonal form αij = α0δij then obviously our
results will not 
hange when we 
hange the geometry of
the latti
e simply be
ause the shunt resistors 
onne
ting
the islands to ground are the same for ea
h island. On
the other hand if we take into 
onsideration that α0 = 0
and only α1 is present, the situation 
hanges be
ause now
values of the dissipative matrix strongly depend on the
Jk spe
trum whi
h indi
ates various values of the matrix
αij depend on the geometry of the stru
ture. This 
ase
is present in arrays with shunt resistors in parallel to the
jun
tions.
A standard way to study models with only diagonal

harging energies Cij = Cδij 
orresponds to a 
om-
plete absen
e of s
reening by the other islands in the
array.
Noti
e the mutual 
apa
itan
e 
an be at least
two orders of magnitude larger than the self-
apa
itan
e
C1 ≃ 102C0. We propose a more realisti
 model in whi
h
both self and mutual 
apa
itan
es are nonzero. To see
how signi�
ant this 
onsideration is we shall analyze Fig.
3. If we assume, that C1/C0 ≃ 50 (see the lowermost line
in Fig. 3) we 
an see the 
riti
al values of the dissipation
parameters 
hange dramati
ally from lo
al α0 = 2 and
nonlo
al α1 = 1 (Appendix A) obtained for fewer real-
isti
 
ases C1/C0 = 1 in whi
h both of the 
apa
itan
es
are 
omparable. In the limit (C1 6= 0, C0 = 0) the latti
e
model is equivalent to the Coulomb gas model with 
rit-
i
al properties not fully understood at present. However
we have to emphasize that the range of the Coulomb ma-
trix be
omes in�nite when C0 is set equal to zero. The
phase boundary in Fig. 3 shift downward with in
reasing
ratio C1/C0 be
ause lower value of the Coulomb inter-
a
tion between nearest neighbors redu
es strong quan-
tum phase �u
tuations and in 
onsequen
e we observe a
growth of long-range phase 
oheren
e.
Yagi et al.
experimented with the super
ondu
tor-
insulator transition in two-dimensional network of
Josephson jun
tions in detail by varying the jun
tions-
area. It was observed the 
riti
al tunneling resistan
e
exhibited signi�
ant jun
tion area dependen
e. The
low-temperature behavior of the total 
harging energy
EJ/EC as a fun
tion RQ/Rn, where Rn is the tunnel-
ing resistan
e exhibits the same behavior as the 
urve
obtained from our theory for square latti
es (see Figure
1) in the absen
e of the lo
al dissipation e�e
ts α0 = 0.
The observed phase boundary, whi
h is bending down-
ward, and the 
riti
al value of the nonlo
al dissipation
parameter αcrit1 = 1 is in ex
ellent a

ordan
e with our
results.
In order to investigate the e�e
ts of quantum �u
tu-
ations and dissipation in JJA's, another group
two-dimensional arrays of small jun
tions with various
EJ/EC and resistors whi
h 
aused dissipation e�e
ts.
The value of EJ was 
ontrolled by varying the tunnel
resistan
e. Ea
h island was 
onne
ted to the neighbor-
ing ones by a shunt resistor as well as the tunnel jun
-
tion. The resistan
e of the shunt resistors was tuned by
varying their length. Ground states of 2D Josephson ar-
ray in EJ/EC -RQ/Rs parameters spa
e reveal the same
behavior as previous experimental results but there is a
di�eren
e in 
riti
al value of the RQ/Rs ≃ 0.5 whi
h also
di�ers from value αcrit1 obtained in this paper. These dis-

repan
ies between experiments 
an be explained in the
framework of our model by taking into a

ount di�er-
ent value of the ratio jun
tion-to-self 
apa
itan
es whi
h
redu
es the 
riti
al value of the RQ/Rs and 
onsidering
that not only nonlo
al dissipation me
hanism is present
α0 6= 0 we 
an obtain RQ/Rs ≃ 0.5 value (see Figure 3).
Real numbers strongly depend on the properties of the
jun
tions used in experiments.
V. SUMMARY
We have 
al
ulated quantum phase diagrams of two-
dimensional Josephson jun
tion arrays using the spher-
i
al model approximation. The 
al
ulations were per-
formed for systems using experimentally attainable ge-
ometries for the arrays su
h us square, triangular and
honey
omb. The ground state of the Josephson 
oupled
array with a triangular latti
e appears to be most stable
against the Coulomb e�e
ts. This geometry is also the

ase in whi
h the global 
oherent state emerged when
the value of the nonlo
al dissipation parameter α1 is the
lowest. In JJA's we 
an observe the phase 
oheren
e
transition whi
h is 
aused by ele
trostati
 and dissipa-
tive e�e
ts. The detailed phase diagrams 
ru
ially de-
pend on the ratio jun
tion-to-self 
apa
itan
es, C1/C0
and both dissipation me
hanism have a big impa
t on
phase boundaries. The nondiagonal terms in 
apa
itive
and dissipative matri
es 
an 
hange the phase diagrams
of the system drasti
ally . It is ne
essary to take them
into 
onsiderations when we have di�erent sour
es of dis-
sipation su
h us shunt resistors 
onne
ting the islands to
a ground and shunt resistors in parallel to the jun
tions.
The experimental observation of an universal resistan
e
threshold for the onset of the global 
oherent state seems
possible, but appears to be di�
ult.
A
knowledgments
One of the authors wants to thank Dr. Ettore Sarnelli
for 
arefully reading manus
ript and fruitful dis
ussions.
This work was supported by the TRN �DeQUACS� and
some parts of it were done in Max-Plan
k-Institut für
DOS △ H �
Jmax/EJ 3
Table I: Maximum values of the spe
trum J (k) for three dif-
ferent geometries of the latti
es: triangular (△), honey
omb
(H) and square (�)
Physik komplexer Systeme, Nöthnitzer Straÿe 38, 01187
Dresden, Germany.
Appendix A: SOME PROPERTIES OF THE
CORRELATOR
Assume that α0 = 0 we write expression for the phase-
phase 
orrelation fun
tion (similar to equation used in a
previous 
al
ulations
but modi�ed by dissipative ma-
trix) in form:
W (τ) = exp
1− cos (ωnτ)
ω2n +
. (A1)
It is easy to see the sum over ωn is symmetri
 when we

hange ωn → −ωn. The key to obtain the solution is
a 
al
ulation the sum or the integral under the expo-
nent in Eq. (A1). Be
ause we are going to investigate
low-temperature properties of the 
orrelation fun
tion we

ould write
dω. In that 
ase (getting
rid of abs) for large value τ we write
W (τ) = exp
1− cos (τω)
ω2 + α1
≃ exp
−2γEJ
α1JkEC
)2EJ/α1Jk
where γ = 0.57721 is the Euler-Mas
heroni 
onstant.
Finally, after Fourier transform we see that 
orrelator
W−1 (ωm) ∼ |ωm|2EJ/α1Jmax−1 at zero temperature di-
verges for α1 ≥ 2EJ/Jmax. Quantity Jmax/EJ means
the maximum value of the Jk whi
h di�ers for various
latti
es (see Table I).
Appendix B: DISSIPATION PARAMETER FOR
CONSIDERED LATTICES
In this appendix we give the expli
it formulas for the
dissipation parameter dis
ussed in Se
. II and III.
1. Square latti
e
π (α0 + 4α1)
α0 + 4α1
where
K (x) =
∫ π/2
1− x2 sin2 φ
, (B2)
is the ellipti
 integral of the �rst kind
and the unit step
fun
tion is de�ned by:
Θ(x) =
1 for x > 0
0 for x ≤ 0
. (B3)
For small values of the α1 we 
an write dissipation pa-
rameter for square latti
e as:
α� = α0 + 3α1 −
, (B4)
for large values values of the α1:
3− 2
2. Triangular latti
e
α−1△ =
K (κ) (B6)
where
(2t+ 3)
1/2 − 1
]3/2 [
(2t+ 3)
4 (2t+ 3)
(2t+ 3)
1/2 − 1
]3/2 [
(2t+ 3)
with t = (α0 + 6α1) /2α1.
3. Honey
omb latti
e
α−1H =
α0 + 3α1
K (κ) (B9)
where
(2t− 1)3/2 (2t+ 3)1/2
(B10)
41/4 (2t)
(2t− 1)3/2 (2t+ 3)1/2
(B11)
with t = (α0 + 3α1) /2α1.
Appendix C: DOS FOR CONSIDERED LATTICES
In this appendix we give the expli
it formulas for the
density of states dis
ussed in Se
. II and III.
1. Square latti
e
ρ� (ξ) =
, (C1)
2. Triangular latti
e
ρ△ (ξ) =
−Θ(ξ − 3)
where
3 + 2
3 + 2ξ − ξ2
−Θ(ξ + 1)
3 + 2ξ [Θ (ξ + 1)−Θ(ξ − 3)] , (C3)
κ1 = 4
3 + 2ξ
−Θ(ξ + 1)
3 + 2
3 + 2ξ − ξ2
[Θ (ξ + 1)−Θ(ξ − 3)] .(C4)
3. Honey
omb latti
e
ρH (ξ) = 4 |ξ| ρ△
3− 4ξ2
. (C5)
Ele
troni
 address: polak��si
a.
ib.na.
nr.it
Ele
troni
 address: kope
�int.pan.wro
.pl
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mailto:polak@fisica.cib.na.cnr.it
mailto:kopec@int.pan.wroc.pl
