- Email: [email protected]
Reversible to Irreversible Flow Transition of Periodically Driven Vortices in the Strip Sample
Available online at www.sciencedirect.com
ScienceDirect Physics Procedia 65 (2015) 105 – 108
27th International Symposium on Superconductivity, ISS 2014
Reversible to irreversible flow transition of periodically driven vortices in the strip sample R. Nitta, Y. Kawamura, S. Kaneko, S. Okuma* Department of Physics, Tokyo Institute of Technology, 2-12-1 Ohokayama, Meguro-ku, Tokyo 152-8551 Japan
Abstract The reversible to irreversible flow transition (RIT) has been observed in periodically driven colloidal suspensions and superconducting vortices in a Corbino-disk (CD) where a global shear is applied. To examine whether RIT is also observed in a system without the global shear but with a local shear, we have studied a vortex system of a strip-shaped amorphous MoxGe1-x film in which only the local shear due to strong random pinning is present. We again obtain evidence of RIT with a critical behavior similar to that for CD. However, the reversible phase is suppressed and the relaxation time for the system to settle into the steady state grows significantly, whose origin is attributed to the stronger pinning effects. © Published by by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license © 2015 2015The TheAuthors. Authors. Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the ISS 2014 Program Committee. Peer-review under responsibility of the ISS 2014 Program Committee Keywords: nonequilibrium transition; vortex; noise; amorphous film; absorbing transition; random organization
1. Introduction We have shown in our recent work that a reversible to irreversible flow transition (RIT), as originally observed in periodically sheared colloidal suspensions [1, 2], also occurs in periodically sheared vortices in a Corbino disk (CD) of an amorphous (a-)MoxGe1-x film [3,4]. In both systems, the particles (vortices) are rotated back and forth around the center of the system (sample) by the global shear force. It has been predicted numerically that for the periodically sheared vortex system, even though we use the strip samples where the global shear is absent, RIT should be observed as long as the samples contain moderately strong pinning that generates local shear [5]. As far as we know, however, this prediction has not been directly proved by experiments. The purpose of this work is to verify this prediction using the same material system where the static and dynamic properties of vortices are similar to each other. We have prepared a strip-shaped film of a-MoxGe1-x with stronger pinning than that for CD studied previously [3,4] to induce the substantial local shear. Then, we have measured flow noise SV and time evolution of the voltage V(t) generated by vortices subjected to ac drive. We again obtain evidence of RIT with a critical behavior similar to that for CD. However, the reversible phase is much suppressed and the relaxation time for the system to settle into the steady state grows significantly. They are attributed to the stronger pinning effects in the present strip sample.
* Corresponding author. Tel.: +81+3-5734-3252; fax: +81+3-5734-2749. E-mail address: [email protected]
1875-3892 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the ISS 2014 Program Committee doi:10.1016/j.phpro.2015.05.145
106
R. Nitta et al. / Physics Procedia 65 (2015) 105 – 108
2. Experimental The a-MoxGe1-x film with thickness of 310 nm was prepared by rf sputtering on a silicon substrate mounted on a water cooled rotating copper stage. The superconducting transition temperature at which the resistivity vanishes is 6.4 K. The pinning strength in the present sample, as estimated from the depinning current density, is about one order of magnitude larger than that for the previous CD sample [3,4]. The resistivity was measured using a standard four-terminal method. We also measured the time evolution of voltage V(t) just after the ac current Iac of square waveform was applied to the vortex system, where the amplitude of Iac was fixed to yield ac voltage with constant amplitude Vf . In measuring noise spectra SV(f) over a broad frequency f range (1 Hz-40 kHz), voltage enhanced with a preamplifier was analyzed with a fast-Fourier transform spectrum analyzer [3,4]. We obtained excess noise spectra by subtracting the background contribution, which was measured with zero current. The sample was directly immersed into the liquid 4He. The magnetic field B was applied perpendicular to the plane of the film. 3. Results and discussion
12
2
SV(100Hz) (aV /Hz)
All the data were taken at 4.1 K in 4.0 T corresponding to the coexistence regime of ordered and disordered phases at equilibrium, where large flow noise appears [6]. We applied Iac of square wave form, whose amplitude was fixed to yield ac voltage with amplitude Vf | 10 P V in the steady state (at t o f ). For 10 P V the vortex flow state is pinning-dominated plastic flow. The period tac of Iac was varied from 0.05 to 20 ms to change the displacement d per half a cycle in the range 0.1-30 P m. Here, d is calculated from the relation d= Vf tac /2Bl, where l is a distance between voltage contacts.
8 4
W (cycles)
0
1
W (cycles)
0
1500
(a)
1000
10
3
10
2
10
1
2
Q =1.3
0
10 −2 10
10
500
−1
10
0
10
1
|d−dc| (Pm)
(e) 0 0
dc=0.1 Pm
1
2
d (Pm)
Fig. 1. (a) SV(100Hz) at 4.1 K in 4.0 T vs d. Data points below the background level (a horizontal dashed line) are indicated with open circles. A full line is a linear fit of the data above the background level (solid circles). V(t) at 4.1 K in 4.0 T for (b) d=1.49 P m, (c) 0.58 P m, and (d) 0.17 P m plotted against the number of drive cycles. Full lines represent the fit of |V(t)| to eq. 1 and horizontal dashed lines indicate the steady-state value of |V(t)|. (e) W vs d. A full line is a power-law fit and a vertical dashed line marks the location of dc. Inset: The same data and its fit plotted on a loglog scale.
R. Nitta et al. / Physics Procedia 65 (2015) 105 – 108
Spectral shape of SV is of Lorentzian type on which narrowband noise originating from the fundamental and higher order frequencies of Iac are superimposed. For small d, some data points are missing from the spectra, indicating that SV is close to the background level. On the other hand, for larger d, substantial broad-band noise exceeding the background level appears. In Fig. 1(a) we plot the d dependence of SV at low frequency (f =100 Hz). For d smaller than about 0.3 P m, SV(100Hz) (open circles) is below the background level (indicated with a dashed line). As d exceeds about 0.5 P m, SV(100Hz) (solid circles) starts to rise and increases almost linearly with d. The curve of SV vs d shown in Fig. 1(a) is qualitatively similar to that observed in the CD sample [3,4]. A threshold value of d where SV disappears is determined to be d0= 0.15 r 0.07 P m from the simple linear extrapolation of the data (solid circles) to the abscissa (SV =0). We tentatively interpret d0 as the threshold displacement of RIT. We note that in the present sample the absolute value of d0= 0.15 P m is about an order of magnitude smaller than dc= 1.6 P m obtained in the CD sample. Hence the relative error is much larger, which makes the determination of dc less accurate. Let us focus on the transient behavior in response to the ac current Iac. Figures 1(b), 1 (c), and 1 (d), respectively, display V(t) versus the number of drive cycles measured for d =1.49, 0.58, and 0.17 P m above and at around d0 (= 0. 15
r 0.07 P m). For each measurement, in order to prepare a disordered initial vortex configuration, we first drive the vortex system by applying dc current yielding the voltage of 100 P V, corresponding to the plastic flow regime, for more than 60 s and then freeze the vortex configuration by turning off the current abruptly [4,7,8]. It is evident from Figs. 1(b)-1(d) that for each d the amplitude of voltage |V(t)|( { V0) at t=0 is smaller than that of |V(t)|( { Vf ) at t o f . This is because the initial vortex distributions are disordered and hence the first several cycles [Fig. 1(b)] to hundreds of cycles [Fig. 1(d)] generate many collisions, similarly to the case of colloids [1,2]. After the many cycles, whose number is largely dependent on d, |V(t)| increases and eventually reaches the steady-state value Vf | 10 P V. For d < 0.1 P m, we were not able to observe the transient behaviour within our experimental resolutions. According to the picture of RIT, for d < d0, which corresponds to the reversible state, all the vortices eventually find a position such that they no longer collide with each other when the system is sheared. On the other hand, for d > d0, which corresponds to the irreversible state, the system reaches the steady state where nonzero fraction of vortices always collides with another vortex [3,4]. Now, we examine the characteristic time W for the system to reach the steady state. W is extracted by fitting the amplitude of the ac voltage |V(t)| to the simple relaxation function [1-5,9],
V (t )
Vf (Vf V0 ) exp( t / W ) / t D .
(1)
Here, we fix D to be zero, because the theory predicts that the value of D becomes relevant only very close to the transition ( W o f ), while in our experiment W / t is relatively small. In Fig. 1(e) we plot W against d. For d < 0.1 P m, we cannot obtain reliable data because of limitation of fast V(t) measurements. The divergence in W is clearly visible at around 0.1 P m, which is close to d0 determined from noise SV(d) in Fig. 1(a). This fact strongly supports the view that the RIT, analogous to the absorbing transition [1-5,7-21], from nonfluctuating (reversible) to fluctuating (irreversible) states occurs at around 0.1 P m ( { dc). The W vs d curve near the transition can be fit to a power-law form, W v | d- dc |-Q with Q =1.3, as shown with a full line. The inset displays the same data plotted on a log-log scale, i.e., log W vs log d d c , giving Q =1.3 r 0.15. This exponent is close to
1.1 r 0.3, 1.33 r 0.02, and 1.3 r 0.3 reported in the colloidal experiments, simulations [1,2], and experiments on vortices in CD [3,4], respectively. From these results, we conclude that evidence of RIT with a critical behavior similar to that for CD is also obtained in the strip sample where the global shear is absent. Our results are consistent with the theoretical prediction that [5] the random pinning centers in superconductors create velocity dispersion by generating local shear in the vortex lattice, which gives rise to the RIT. Compared to the results for CD, the reversible phase is much suppressed down to near zero (d < 0.1 P m) and the relaxation time for the system to settle into the steady state grows significantly from several cycles for CD [3,4] to a thousand of cycles in the present strip-shaped sample. Their origins are attributed to the stronger pinning effects in the present sample. Owing to the significantly increased values of W , the accuracy of Q (=1.3 r 0.15) extracted from the power-law fit of W(d ) is improved. 4. Summary
To summarize, the non-equilibrium transition, RIT, has been observed in periodically driven colloidal suspensions [1,2] and vortices in CD [3,4] where the global shear is present. However, it has not yet been clarified experimentally whether the RIT is also observed in a system where the global shear is absent, while the theory has answered it
107
108
R. Nitta et al. / Physics Procedia 65 (2015) 105 – 108
affirmatively [5]. To clarify the issue, in this work we have studied a vortex system of a strip-shaped a-MoxGe1-x film in which only local shear resulting from random pinning centers is present, but the pinning strength is stronger than that for CD studied previously [3,4]. We again obtain evidence of RIT with a critical behavior similar to that for CD, consistent with the theoretical prediction [5]. However, the reversible phase is much suppressed and the relaxation time increases significantly, whose origins are attributed to stronger pinning effects. Acknowledgements We thank C. Reichhardt and K.A. Takeuchi for helpful discussions. Y. K. acknowledges the financial support from the Global Center of Excellence Program by MEXT, Japan through the "Nanoscience and Quantum Physics" Project of the Tokyo Institute of Technology. This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. References [1] D. J. Pine, J. P. Gollub, J.F. Brady, A.M. Leshansky, Nature 438 (2005) 997. [2] J. Gollub, D. Pine, Phys. Today 59 (2006) 8. [3] S. Okuma, Y. Tsugawa, A. Motohashi, Phys. Rev. B 83 (2011) 012503 [4] S. Okuma, Y. Kawamura, Y. Tsugawa, J. Phys. Soc. Jpn. 81 (2012) 114718. [5] N. Mangan, C. Reichhardt, C.J. Olson Reichhardt, Phys. Rev. Lett. 100 (2008) 187002. [6] S. Okuma, K. Suzuki, Y. Suzuki, N. Kokubo, Phys. Rev. B 77 (2008) 212505. [7] S. Okuma, A. Motohashi, New J. Phys. 14 (2012) 123021. [8] S. Okuma, A. Motohashi, Y. Kawamura, Phys. Lett. A 377 (2013) 2990. [9] C. Reichhardt, C. J. Olson Reichhardt, Phys. Rev. Lett. 103 (2009) 168301. [10] H. Hinrichsen, Adv. Phys. 49 (2000) 815. [11] K.A. Takeuchi, M. Kuroda, H. Chate, M. Sano, Phys. Rev. Lett. 99 (2007) 234503. [12] W. Zhang, W. Zhou, M. Luo, Phys. Lett. A 374 (2010) 3666. [13] Hatem Barghathi, Thomas Vojta, Phys. Rev. Lett. 109 (2012) 170603. [14] C. Zhou, C.J. Olson Reichhardt, C. Reichhardt, I. Beyerlein, Phys. Lett. A 378 (2014) 1675. [15] Kazumasa A. Takeuchi, Masafumi Kuroda, Hugues Chaté, Masaki Sano, Phys. Rev. E 80 (2009) 051116. [16] C. Reichhardt, C. J. O. Reichhardt, Proc. Natl. Acad. Sci. (USA) 108 (2011) 19099. [17] Kazumasa A Takeuchi, J. Stat. Mech. (2014) P01006. [18] Thomas Vojta, Phys. Rev. E 86 (2012) 051137. [19] Raphael Jeanneret, Denis Bartolo, Nat. Comms. 5 (2014) 3474. [20] Hatem Barghathi, David Nozadze, Thomas Vojta, Phys. Rev. E 89 (2014) 012112. [21] Marcelo Martins de Oliveira, Ronald Dickman, Phys. Rev. E 90 (2014) 032120.
ScienceDirect Physics Procedia 65 (2015) 105 – 108
27th International Symposium on Superconductivity, ISS 2014
Reversible to irreversible flow transition of periodically driven vortices in the strip sample R. Nitta, Y. Kawamura, S. Kaneko, S. Okuma* Department of Physics, Tokyo Institute of Technology, 2-12-1 Ohokayama, Meguro-ku, Tokyo 152-8551 Japan
Abstract The reversible to irreversible flow transition (RIT) has been observed in periodically driven colloidal suspensions and superconducting vortices in a Corbino-disk (CD) where a global shear is applied. To examine whether RIT is also observed in a system without the global shear but with a local shear, we have studied a vortex system of a strip-shaped amorphous MoxGe1-x film in which only the local shear due to strong random pinning is present. We again obtain evidence of RIT with a critical behavior similar to that for CD. However, the reversible phase is suppressed and the relaxation time for the system to settle into the steady state grows significantly, whose origin is attributed to the stronger pinning effects. © Published by by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license © 2015 2015The TheAuthors. Authors. Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the ISS 2014 Program Committee. Peer-review under responsibility of the ISS 2014 Program Committee Keywords: nonequilibrium transition; vortex; noise; amorphous film; absorbing transition; random organization
1. Introduction We have shown in our recent work that a reversible to irreversible flow transition (RIT), as originally observed in periodically sheared colloidal suspensions [1, 2], also occurs in periodically sheared vortices in a Corbino disk (CD) of an amorphous (a-)MoxGe1-x film [3,4]. In both systems, the particles (vortices) are rotated back and forth around the center of the system (sample) by the global shear force. It has been predicted numerically that for the periodically sheared vortex system, even though we use the strip samples where the global shear is absent, RIT should be observed as long as the samples contain moderately strong pinning that generates local shear [5]. As far as we know, however, this prediction has not been directly proved by experiments. The purpose of this work is to verify this prediction using the same material system where the static and dynamic properties of vortices are similar to each other. We have prepared a strip-shaped film of a-MoxGe1-x with stronger pinning than that for CD studied previously [3,4] to induce the substantial local shear. Then, we have measured flow noise SV and time evolution of the voltage V(t) generated by vortices subjected to ac drive. We again obtain evidence of RIT with a critical behavior similar to that for CD. However, the reversible phase is much suppressed and the relaxation time for the system to settle into the steady state grows significantly. They are attributed to the stronger pinning effects in the present strip sample.
* Corresponding author. Tel.: +81+3-5734-3252; fax: +81+3-5734-2749. E-mail address: [email protected]
1875-3892 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the ISS 2014 Program Committee doi:10.1016/j.phpro.2015.05.145
106
R. Nitta et al. / Physics Procedia 65 (2015) 105 – 108
2. Experimental The a-MoxGe1-x film with thickness of 310 nm was prepared by rf sputtering on a silicon substrate mounted on a water cooled rotating copper stage. The superconducting transition temperature at which the resistivity vanishes is 6.4 K. The pinning strength in the present sample, as estimated from the depinning current density, is about one order of magnitude larger than that for the previous CD sample [3,4]. The resistivity was measured using a standard four-terminal method. We also measured the time evolution of voltage V(t) just after the ac current Iac of square waveform was applied to the vortex system, where the amplitude of Iac was fixed to yield ac voltage with constant amplitude Vf . In measuring noise spectra SV(f) over a broad frequency f range (1 Hz-40 kHz), voltage enhanced with a preamplifier was analyzed with a fast-Fourier transform spectrum analyzer [3,4]. We obtained excess noise spectra by subtracting the background contribution, which was measured with zero current. The sample was directly immersed into the liquid 4He. The magnetic field B was applied perpendicular to the plane of the film. 3. Results and discussion
12
2
SV(100Hz) (aV /Hz)
All the data were taken at 4.1 K in 4.0 T corresponding to the coexistence regime of ordered and disordered phases at equilibrium, where large flow noise appears [6]. We applied Iac of square wave form, whose amplitude was fixed to yield ac voltage with amplitude Vf | 10 P V in the steady state (at t o f ). For 10 P V the vortex flow state is pinning-dominated plastic flow. The period tac of Iac was varied from 0.05 to 20 ms to change the displacement d per half a cycle in the range 0.1-30 P m. Here, d is calculated from the relation d= Vf tac /2Bl, where l is a distance between voltage contacts.
8 4
W (cycles)
0
1
W (cycles)
0
1500
(a)
1000
10
3
10
2
10
1
2
Q =1.3
0
10 −2 10
10
500
−1
10
0
10
1
|d−dc| (Pm)
(e) 0 0
dc=0.1 Pm
1
2
d (Pm)
Fig. 1. (a) SV(100Hz) at 4.1 K in 4.0 T vs d. Data points below the background level (a horizontal dashed line) are indicated with open circles. A full line is a linear fit of the data above the background level (solid circles). V(t) at 4.1 K in 4.0 T for (b) d=1.49 P m, (c) 0.58 P m, and (d) 0.17 P m plotted against the number of drive cycles. Full lines represent the fit of |V(t)| to eq. 1 and horizontal dashed lines indicate the steady-state value of |V(t)|. (e) W vs d. A full line is a power-law fit and a vertical dashed line marks the location of dc. Inset: The same data and its fit plotted on a loglog scale.
R. Nitta et al. / Physics Procedia 65 (2015) 105 – 108
Spectral shape of SV is of Lorentzian type on which narrowband noise originating from the fundamental and higher order frequencies of Iac are superimposed. For small d, some data points are missing from the spectra, indicating that SV is close to the background level. On the other hand, for larger d, substantial broad-band noise exceeding the background level appears. In Fig. 1(a) we plot the d dependence of SV at low frequency (f =100 Hz). For d smaller than about 0.3 P m, SV(100Hz) (open circles) is below the background level (indicated with a dashed line). As d exceeds about 0.5 P m, SV(100Hz) (solid circles) starts to rise and increases almost linearly with d. The curve of SV vs d shown in Fig. 1(a) is qualitatively similar to that observed in the CD sample [3,4]. A threshold value of d where SV disappears is determined to be d0= 0.15 r 0.07 P m from the simple linear extrapolation of the data (solid circles) to the abscissa (SV =0). We tentatively interpret d0 as the threshold displacement of RIT. We note that in the present sample the absolute value of d0= 0.15 P m is about an order of magnitude smaller than dc= 1.6 P m obtained in the CD sample. Hence the relative error is much larger, which makes the determination of dc less accurate. Let us focus on the transient behavior in response to the ac current Iac. Figures 1(b), 1 (c), and 1 (d), respectively, display V(t) versus the number of drive cycles measured for d =1.49, 0.58, and 0.17 P m above and at around d0 (= 0. 15
r 0.07 P m). For each measurement, in order to prepare a disordered initial vortex configuration, we first drive the vortex system by applying dc current yielding the voltage of 100 P V, corresponding to the plastic flow regime, for more than 60 s and then freeze the vortex configuration by turning off the current abruptly [4,7,8]. It is evident from Figs. 1(b)-1(d) that for each d the amplitude of voltage |V(t)|( { V0) at t=0 is smaller than that of |V(t)|( { Vf ) at t o f . This is because the initial vortex distributions are disordered and hence the first several cycles [Fig. 1(b)] to hundreds of cycles [Fig. 1(d)] generate many collisions, similarly to the case of colloids [1,2]. After the many cycles, whose number is largely dependent on d, |V(t)| increases and eventually reaches the steady-state value Vf | 10 P V. For d < 0.1 P m, we were not able to observe the transient behaviour within our experimental resolutions. According to the picture of RIT, for d < d0, which corresponds to the reversible state, all the vortices eventually find a position such that they no longer collide with each other when the system is sheared. On the other hand, for d > d0, which corresponds to the irreversible state, the system reaches the steady state where nonzero fraction of vortices always collides with another vortex [3,4]. Now, we examine the characteristic time W for the system to reach the steady state. W is extracted by fitting the amplitude of the ac voltage |V(t)| to the simple relaxation function [1-5,9],
V (t )
Vf (Vf V0 ) exp( t / W ) / t D .
(1)
Here, we fix D to be zero, because the theory predicts that the value of D becomes relevant only very close to the transition ( W o f ), while in our experiment W / t is relatively small. In Fig. 1(e) we plot W against d. For d < 0.1 P m, we cannot obtain reliable data because of limitation of fast V(t) measurements. The divergence in W is clearly visible at around 0.1 P m, which is close to d0 determined from noise SV(d) in Fig. 1(a). This fact strongly supports the view that the RIT, analogous to the absorbing transition [1-5,7-21], from nonfluctuating (reversible) to fluctuating (irreversible) states occurs at around 0.1 P m ( { dc). The W vs d curve near the transition can be fit to a power-law form, W v | d- dc |-Q with Q =1.3, as shown with a full line. The inset displays the same data plotted on a log-log scale, i.e., log W vs log d d c , giving Q =1.3 r 0.15. This exponent is close to
1.1 r 0.3, 1.33 r 0.02, and 1.3 r 0.3 reported in the colloidal experiments, simulations [1,2], and experiments on vortices in CD [3,4], respectively. From these results, we conclude that evidence of RIT with a critical behavior similar to that for CD is also obtained in the strip sample where the global shear is absent. Our results are consistent with the theoretical prediction that [5] the random pinning centers in superconductors create velocity dispersion by generating local shear in the vortex lattice, which gives rise to the RIT. Compared to the results for CD, the reversible phase is much suppressed down to near zero (d < 0.1 P m) and the relaxation time for the system to settle into the steady state grows significantly from several cycles for CD [3,4] to a thousand of cycles in the present strip-shaped sample. Their origins are attributed to the stronger pinning effects in the present sample. Owing to the significantly increased values of W , the accuracy of Q (=1.3 r 0.15) extracted from the power-law fit of W(d ) is improved. 4. Summary
To summarize, the non-equilibrium transition, RIT, has been observed in periodically driven colloidal suspensions [1,2] and vortices in CD [3,4] where the global shear is present. However, it has not yet been clarified experimentally whether the RIT is also observed in a system where the global shear is absent, while the theory has answered it
107
108
R. Nitta et al. / Physics Procedia 65 (2015) 105 – 108
affirmatively [5]. To clarify the issue, in this work we have studied a vortex system of a strip-shaped a-MoxGe1-x film in which only local shear resulting from random pinning centers is present, but the pinning strength is stronger than that for CD studied previously [3,4]. We again obtain evidence of RIT with a critical behavior similar to that for CD, consistent with the theoretical prediction [5]. However, the reversible phase is much suppressed and the relaxation time increases significantly, whose origins are attributed to stronger pinning effects. Acknowledgements We thank C. Reichhardt and K.A. Takeuchi for helpful discussions. Y. K. acknowledges the financial support from the Global Center of Excellence Program by MEXT, Japan through the "Nanoscience and Quantum Physics" Project of the Tokyo Institute of Technology. This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. References [1] D. J. Pine, J. P. Gollub, J.F. Brady, A.M. Leshansky, Nature 438 (2005) 997. [2] J. Gollub, D. Pine, Phys. Today 59 (2006) 8. [3] S. Okuma, Y. Tsugawa, A. Motohashi, Phys. Rev. B 83 (2011) 012503 [4] S. Okuma, Y. Kawamura, Y. Tsugawa, J. Phys. Soc. Jpn. 81 (2012) 114718. [5] N. Mangan, C. Reichhardt, C.J. Olson Reichhardt, Phys. Rev. Lett. 100 (2008) 187002. [6] S. Okuma, K. Suzuki, Y. Suzuki, N. Kokubo, Phys. Rev. B 77 (2008) 212505. [7] S. Okuma, A. Motohashi, New J. Phys. 14 (2012) 123021. [8] S. Okuma, A. Motohashi, Y. Kawamura, Phys. Lett. A 377 (2013) 2990. [9] C. Reichhardt, C. J. Olson Reichhardt, Phys. Rev. Lett. 103 (2009) 168301. [10] H. Hinrichsen, Adv. Phys. 49 (2000) 815. [11] K.A. Takeuchi, M. Kuroda, H. Chate, M. Sano, Phys. Rev. Lett. 99 (2007) 234503. [12] W. Zhang, W. Zhou, M. Luo, Phys. Lett. A 374 (2010) 3666. [13] Hatem Barghathi, Thomas Vojta, Phys. Rev. Lett. 109 (2012) 170603. [14] C. Zhou, C.J. Olson Reichhardt, C. Reichhardt, I. Beyerlein, Phys. Lett. A 378 (2014) 1675. [15] Kazumasa A. Takeuchi, Masafumi Kuroda, Hugues Chaté, Masaki Sano, Phys. Rev. E 80 (2009) 051116. [16] C. Reichhardt, C. J. O. Reichhardt, Proc. Natl. Acad. Sci. (USA) 108 (2011) 19099. [17] Kazumasa A Takeuchi, J. Stat. Mech. (2014) P01006. [18] Thomas Vojta, Phys. Rev. E 86 (2012) 051137. [19] Raphael Jeanneret, Denis Bartolo, Nat. Comms. 5 (2014) 3474. [20] Hatem Barghathi, David Nozadze, Thomas Vojta, Phys. Rev. E 89 (2014) 012112. [21] Marcelo Martins de Oliveira, Ronald Dickman, Phys. Rev. E 90 (2014) 032120.