PHYSICAL REVIEW B 80, 094502 共2009兲

Duality breaking of vortex configuration in a hierarchical honeycomb network
S. Tsuchiya,1 T. Toshima,2 H. Nobukane,3 K. Inagaki,1,4 and S. Tanda1,41Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
2Department of Ecomaterials Engineering, Toyama National College of Technology, Toyama 939-8630, Japan
3Department of Physics, Asahikawa Medical College, Asahikawa, Hokkaido 078-8510, Japan
4Center of Education and Research for Topological Science and Technology, Hokkaido University, Sapporo 060-8628, Japan

共Received 23 June 2009; revised manuscript received 11 August 2009; published 3 September 2009兲
We report measurements of Little-Parks oscillation on the hierarchical honeycomb-superconducting network
for investigating possible effects of hierarchical structure in terms of spatial symmetry, parity, and duality. Weobserved an asymmetric Little-Parks oscillation about ⌽ / ⌽0 = 1 / 2, although spatial symmetry was kept in thenetwork. In comparison with a regular honeycomb network, the asymmetric oscillation is attributed to hierarchy which induces mixture of commensurate and incommensurate regions. The asymmetric oscillation is foundto indicate breaking of the duality of vortex configuration.
DOI: 10.1103/PhysRevB.80.094502

PACS number共s兲: 74.81.Fa, 64.60.aq, 02.40.Tt

I. INTRODUCTION

Complexity, more specifically, hierarchical structure has
attracted attention to the problem of decoherence1 leading to
the quantum computing2 and the so-called quantum-toclassical transition of cosmological perturbations in the early
universe that grew up to galaxies and clusters of galaxies.3–5
Since the hierarchical structure is found in the large-scale
structure of galaxies, it may play an important role for thegeneration of the density fluctuation in the early universe.
However, their structural effect on the quantum-to-classicaltransition has remained unclear.
Superconducting network allows us to perform a proper
test for the structural effect of hierarchy. In superconducting
networks, the structural properties well affect the physicalproperties because this system is sensitive to phase coherence of the order parameter over the network. In fact, phaseinterference phenomena are driven by the magnetic field
known as Little-Parks oscillation.6 Characteristic vortex entry and configuration of vortices is caused as a result of thestructural effect.7–10 Therefore one can observe the structuraleffects as dips or cusps of variation in magnetic field responses. In addition, complex structure can be easily designed.
Generally, in a spatially symmetric system such as regularperiodic networks 共square, triangular, and honeycomb lattices兲, the duality on these networks is conserved.11,12 However, in a hierarchical network, it is a nontrivial issue
whether the duality arises or not. Because their periodicity isdifferent from that of the regular network although spatial
symmetry of the hierarchical network is kept. In this paper,
we examine possible effects of a hierarchical structure interms of spatial symmetry, parity, and duality of order parameter by using superconducting networks. Our results,
which were observed as an asymmetric Little-Parks oscillation about ⌽ / ⌽0 = 1 / 2, indicates breaking of the duality ofvortex configuration on the network due to the effect of hierarchical structure.
II. EXPERIMENTAL

For our experiment, the so-called hierarchical honeycomb
structure, which was discovered in two-dimensional charge1098-0121/2009/80共9兲/094502共4兲

density wave system, was adopted.13 In this structure, theelectrons which align on the triangle lattice grow up withhexagonal smoothing clusters hierarchically. The propagation law was applied to superconducting networks. Thisstructure is based on the concept of smoothing and is different from a simple fractal form such as Sielpinski gasket.
The lead networks that we used were fabricated by standard electron-beam lithography. The gold adhesion layer of0.01 ␮m and the lead layer of 0.1 ␮m are thermally evaporated on a SiO2 substrate followed by the resist lift off. To
compare hierarchical structure with regular structure, we also
prepared a regular honeycomb network in the same way.
Figure 1 shows a scanning electron microscope 共SEM兲 image of the samples. The regular sample has about 5000 cellswith lattice constant of 2 ␮m, line width of 0.6 ␮m. In thehierarchical one, the elementary hexagon side length is2 ␮m with line width of 0.2 ␮m and five classes of hierarchy.
Little-Parks oscillation is a powerful tool to investigate
the configuration of vortices on the network. Little-Parks oscillation is a periodic variation in superconducting transition
temperature 共Tc兲 with the magnetic field by the superconducting fluxoid quantization.6 Especially when temperatureis near Tc, phase coherence is stretched over the whole system. Hence variation of Tc is affected by vortex configuration. Experimentally, Little-Parks oscillation of Tc can be

FIG. 1. SEM image of the samples. 共a兲 The regular honeycomb
network, which has about 5000 cells with lattice constant of 2 ␮m,line width of 0.6 ␮m. 共b兲 The hierarchical honeycomb network.
The elementary hexagon side length is 2 ␮m with line width of0.2 ␮m and has five classes of hierarchy.

094502-1©2009 The American Physical Society

PHYSICAL REVIEW B 80, 094502 共2009兲

TSUCHIYA et al.

(a)
0.3

R/RN

R/RN

0.3

Magnetic flux (Gauss)

121

32

3

4352

53

4

0.25

100.2

10

Slope = 2.22 Gauss / Number

00.2

5

Number of flux quantum

0Magnetic flux (Gauss)

00.5

/

10Duality(b)FIG. 2. The magnetic-flux dependence of the sample resistancenormalized by the normal-state resistance RN. We found periodic
dips indicated by the arrows. The inset shows the index number ofdip positions as a function of the magnetic-flux. The slope shows
the period of oscillation as 2.22 G.

10112132343525340(%)

1.3%

observed as a periodic variation in resistance with the magnetic field at fixed temperature, which was taken near themidpoint of normal-to-superconducting transition.
0.2III. RESULTS AND DISCUSSIONFirst the regular honeycomb network was investigated asa control experiment. Figure 2 shows the magnetic-flux dependence of the sample resistance normalized by the normalstate resistance RN. We found periodic dips indicated by thearrows. The inset of Fig. 2 shows the index number of dip
positions as a function of the magnetic flux. The slope shows
the period of oscillation as 2.22 G. The area calculated fromthe period is 9.3 ␮m2 and corresponds to a hexagonal unit
cell enclosed by the center of the wire. This value compares
well to the value 9.6 ␮m2 obtained from SEM observation
with 3% accuracy. Thus the period corresponds to one-flux
quantum ⌽0 = h / 2e per unit cell.
Figure 3共a兲 presents the sample resistance as a function ofthe filling ratio ⌽ / ⌽0, which is the magnetic-flux ⌽ in unitsof the flux quantum ⌽0 per a hexagonal unit cell, in range
from 0 to 1. The arrows indicate dips with the fundamental
filling ratio of 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, and 3/4. Thesefundamental dips appeared in different range, for example,
from 1 to 2, were not shown in the figure. The results seem
like Farey sequence and are consistent with the recent
report.14 Figure 3共b兲 exhibits error from Farey sequence F5= 兵 51 , 41 , 31 , 52 , 21 , 53 , 32 , 43 , 54 其 versus the dip positions. Shape ofsymbols denotes correspondence relation from the viewpointof the symmetry about 1/2. Solid and open symbols denote0 ⬍ ⌽ / ⌽0 ⬍ 1 / 2 and 1 / 2 ⬍ ⌽ / ⌽0 ⬍ 1, respectively. Every
dip correspond to F5 within 1.3% accuracy and are clearly
symmetric about ⌽ / ⌽0 = 1 / 2.
On the other hand, an asymmetric oscillation was observed in the case of the hierarchical honeycomb network.
We found a periodic variation in the magnetoresistance asshown in Figs. 4共a兲 and 4共b兲. Measurement noise has been

0.4

0.6

Dip position

0.8

FIG. 3. 共Color online兲 共a兲 The sample resistance as a function ofthe filling ratio ⌽ / ⌽0 in range from 0 to 1. The arrows indicate dipswith the fundamental filling ratio of 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, and3/4. 共b兲 Error from Farey sequence versus the dip positions. Shapeof symbols denotes correspondence relation in terms of the symmetry about 1/2. Solid and open symbols denote 0 ⬍ ⌽ / ⌽0 ⬍ 1 / 2 and1 / 2 ⬍ ⌽ / ⌽0 ⬍ 1, respectively. Solid line in the center is a guide tothe eye indicating ⌽ / ⌽0 = 1 / 2. Every dips correspond to F5 within
1.3% accuracy. The inset is vortex configuration on the regular
honeycomb network. The unit cells occupied with vortices areshown shaded. Left side is at ⌽ / ⌽0 = 1 / 3 and right side is ⌽ / ⌽0= 2 / 3. Spatial vortex configuration of 2/3, is essentially identical tothat of 1/3.

subtracted. The slope of the line is 2.00 G as shown in Fig.
4共c兲. The area estimated from this period is 10.1 ␮m2 andcorrespond to just about an elementary hexagon in comparison with 9.8 ␮m2 deduced from SEM observation with 3%
accuracy.
Some dips expected from the symmetry about 1/2 were
found to be absent as shown in Fig. 5共a兲. In Fig. 5, measurement noise has been subtracted. Dips indicated by the arrows
do not fully agree with Farey sequence. In Fig. 5共b兲, the dipsof 1/5, 1/3, 1/2, and 3/4 correspond to F5 within 1.3% accuracy, the other dips obviously deviate from F5 with accuracy
up to 6%. This result suggests violation of symmetry about
1/2 in the case of the hierarchical honeycomb network.
Now let us discuss the symmetric and asymmetric oscillation about ⌽ / ⌽0 = 1 / 2 by considering vortex configurationon the network. In the case of the regular honeycomb network, vortices are commensurately arranged with its basestructure at rational ⌽ / ⌽0. For example, at ⌽ / ⌽0 = 1 / 3, one

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PHYSICAL REVIEW B 80, 094502 共2009兲DUALITY BREAKING OF VORTEX CONFIGURATION IN A…

10 (c)(b)

Magnetic flux (Gauss)

0.1

0.1(a)∆R0∆R0

Magnetic flux (Gauss)00

Magnetic flux (Gauss)0

Slope = 2.00 Gauss / Number0105

Number of flux quantum

FIG. 4. The magnetic-flux dependence of the sample resistance normalized by RN. 共a兲 In range from −10 to 0 G. 共b兲 In range from 0 to10 G. The arrows indicate periodic dips. 共c兲 The index number of dip positions as a function of the magnetic flux. The slope of the line is2.00 G.

vortex is allocated in every three unit cell that is shown
shaded in the inset of Fig. 3共b兲. This configuration is strongly
pinned and energetically stable. At ⌽ / ⌽0 = 2 / 3, two vortices
are allocated in every three unit cells in the same way. Whenthe position occupied with vortex is replaced by the position
without vortex, spatial vortex configuration of 2/3, is essen-

R/RN

0.01(a)

01
3
3

1

42

1
50/

10(b)
11112345432345

tially identical to that of 1/3. Both states have energeticallysame eigenvalue. This is a duality of vortex configuration.
Vortex configuration is globally determined at rational magnetic field.
On the other hand, in the case of the hierarchical honeycomb network, the asymmetric oscillation indicates breakingof the duality of vortex configuration although the system
has symmetric structure. To explain the asymmetric oscillation we propose a model of vortex configuration as shown inFig. 6. Blue and red denote commensurate and incommensurate region where vortices are allocated, respectively. Color
depth denotes density of vortices. Figure 6共a兲 shows the configuration at 1/3, where in some regions vortices are commensurately allocated with base structure and other regions
are not, since some different areas exist in the hierarchicalhoneycomb structure. According to the duality of vortex configuration like the regular case 关see the inset of Fig. 3共b兲兴, theconfiguration corresponding to 2/3 is depicted in Fig. 6共b兲. In
spite of duality operation, the configuration of 2/3 is not dualfor that of 1/3. There are no commensurate regions because
the base structure in incommensurate regions at 1/3 and 2/3

1.3%(%)

0CIC

0.2

0.4

0.6

IC(C)
(IC)

0.8

Dip position
FIG. 5. 共Color online兲 共a兲 The sample resistance as a function ofthe filling ratio ⌽ / ⌽0 in range from 0 to 1. The arrows indicate
fundamental dips. 共b兲 Error from Farey sequence versus the dip
positions. Same notation as in Fig. 3 is used. The dashed arrows areguides to the eye indicating correspondence in terms of the symmetry about 1/2. Dips of 1/5, 1/3, 1/2, and 3/4 correspond to F5 within
1.3% accuracy. The other dips deviate from F5 with accuracy up to6%.

FIG. 6. 共Color online兲 The model of vortex configuration on thehierarchical honeycomb network. Blue and red denote commensurate and incommensurate region where vortices are allocated, respectively. Color depth denotes density of vortices. 共a兲 At ⌽ / ⌽0= 1 / 3. In some regions vortices are commensurately allocated withbase structure and other regions are not. 共b兲 At ⌽ / ⌽0 = 2 / 3. In spiteof duality operation, the configuration of 2/3 is not dual for that of1/3.

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TSUCHIYA et al.

is the same. This model could suggest that these states haveenergetically different eigenvalues and explain the asymmetric oscillation about 1/2, in particular, absence of dip at 2/3 inthe hierarchical honeycomb network.
Our result is different from the result of the Sierpinski
gasket.15 As far as this result is concerned, it is found to besymmetric about 1/2. Therefore, the duality of vortex configuration is conserved. Additionally, vortex configuration is
not globally determined at any magnetic field because theircharacteristic length is absent. Hence the asymmetric patternis due to the effect of the hierarchical honeycomb structure in
itself.
Finally we have some comments for general properties ofour hierarchical structure. Strong pinning of vortices means
phase fluctuation of the order parameter is very small or its
phase is determined spatially and temporally. If similar hierarchical structure is existed in some systems, the regions
where its phase is determined could be appeared spontaneously with spatial dependence. This concept might be applied to the problem of decoherence in complex systems.

1W. H. Zurek, Rev. Mod. Phys. 75, 715 共2003兲.

We measured Little-Parks oscillation in the hierarchicalsuperconducting network for investigating effects of hierarchical structure in terms of spatial symmetry, parity, and duality. We observed the asymmetric Little-Parks oscillation
about ⌽ / ⌽0 = 1 / 2, although spatial symmetry was kept in thenetwork. In comparison with a regular honeycomb network,
the asymmetric oscillation is attributed to hierarchy which
induces mixture of commensurate and incommensurate regions. The asymmetric oscillation is found to indicate breaking of the duality of vortex configuration.
ACKNOWLEDGMENTS

The authors are grateful to A. J. Leggett, F. Nori, and K.
Semba for useful discussions. We also thank K. Yamaya, S.
Takayanagi, and N. Matsunaga for experimental support.
This work was supported by the Grant-in-Aid for the 21stCentury COE program “Topological Science and Technology.”

92 S. Aaronson, arXiv:quant-ph/0311039 共unpublished兲.
3 L.

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