Transition by Breaking of Analyticity in the Ground State of Josephson Junction Arrays as a Static Signature of the Vortex Jamming Transition
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y Transition by Breaking of Analyticity in the Ground State of Josephson JunctionArrays as a Static Signature of the Vortex Jamming Transition
Tomoaki Nogawa ∗ Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Hajime Yoshino
Department of Earth and Space Science, Faculty of Science, Osaka University, Toyonaka 560-0043, Japan
Bongsoo Kim
Department of Physics, Changwon National University, Changwon 641-773, Korea (Dated: October 15, 2018)We investigate the ground state of the irrationally frustrated Josephson junction array with con-trolling anisotropy parameter λ that is the ratio of the longitudinal Josephson coupling to thetransverse one. We find that the ground state has one dimensional periodicity whose reciprocallattice vector depends on λ and is incommensurate with the substrate lattice. Approaching theisotropic point, λ =1 the so called hull function of the ground state exhibits analyticity breakingsimilar to the Aubry transition in the Frenkel-Kontorova model. We find a scaling law for theharmonic spectrum of the hull functions, which suggests the existence of a characteristic lengthscale diverging at the isotropic point. This critical behavior is directly connected to the jammingtransition previously observed in the current-voltage characteristics by a numerical simulation. Ontop of the ground state there is a gapless, continuous band of metastable states, which exhibit thesame critical behavior as the ground state. I. INTRODUCTION
Frustration is regarded as a key concept in the under-standing of a number of complex cooperative phenomenain condensed matter systems including the glass transi-tion [1, 2]. In general frustration leads to flattening ofthe energy landscape and suppresses the onset of conven-tional long-range orders, thereby opening possibilities ofexotic phases. In order to understand the roles of frustra-tion, it is desirable to vary the strength of frustration ina systematic way. The frustrated Josephson junction ar-ray (JJA) under an external magnetic field [3] provides anideally simple setting for this purpose since the strengthof the frustration in the JJA can be tuned at will as dis-cussed below. The origin of frustration of this system isthe impossibility to minimize the Josephson junction cou-pling energy of longitudinal and transverse bonds at thesame time when a magnetic field is applied. This mech-anism seems to have a certain generality. The frustratedJJA is closely related to the notion of frustrated crys-tals in curved space [1, 2]. Such frustrated crystals maybe realized, to some extent, by actually bending crystals[4, 5]. This amounts to injecting a given number densityof dislocations into the crystals. Here the key parameterof the frustration is the number density of the externallyinduced dislocations. In the frustrated JJA, the disloca-tions correspond to the vortices induced by an externalmagnetic field. Quite remarkably, one can easily controlthe number density of vortices per plaquette f by justchanging the strength of the applied external magnetic ∗ [email protected] field.By choosing an irrational f instead of a rational f [6]we can maximize the strength of frustration in the sensethat we can suppress the formation of a dislocation (vor-tex) lattice commensurate with respect to the underlyingJJA. This system is called an irrationally frustrated JJA(IFJJA) and has attracted researchers for a long time. Ithas been observed numerically that the system remainsin a vortex liquid state down to low temperatures [7–9]and exhibits glassy signatures [10, 11].Recently it has been found that a relevant and veryinteresting perturbation on the IFJJA is the anisotropy of the Josephson coupling [12–14], which breaks the bal-ance of longitudinal and transverse bonds and thus weak-ens the frustration. Anisotropic JJAs can be fabricatedin laboratories by the lithography technique [15]. TheIFJJA on the square lattice with different strengths ofthe Josephson couplings J x and J y in the x and y di-rections, manifests itself as unusual vortex matter thatslides freely in the direction of stronger coupling evenat zero temperature, similarly to incommensurate chargedensity waves, but jammed into the direction of weakercoupling [12]. It was argued that the mechanism ofthe sliding-jamming transition is similar to that of theFrenkel-Kontorova (FK) model [16].In the present paper, we study how the ground states(GSs) of the IFJJA change with the anisotropy λ = J y /J x . To this end we develop an efficient numericalmethod valid even up to the isotropic point λ = 1, whichsubstantially extends the previous analysis based on a1 /λ expansion [13], whose validity is limited to the stronganisotropy limit λ ≫
1. The GSs are characterized by anincommensurate wave vector q ( λ ) and its higher harmon-ics that continuously vary with λ . With strong enoughanisotropy λ ≫
1, the vortices are aligned as stripes par-allel to the weaker coupling axis [13]. By tuning theanisotropy smaller, the stripes become more tilted withrespect to the weaker coupling (see Fig. 1). We demon-strate that the so called hull function [16], which re-flects the hidden periodic pattern that is incommensu-rate with the substrate lattice, becomes nonanalytic atthe isotropic point λ = 1. Physically this means thatthe sliding becomes prohibited in both the x and y direc-tions at the isotropic point, leading to a jamming tran-sition, which is analogous to the so-called Aubry tran-sition, which is well known in the FK model [16]. Thisanalyticity-breaking transition is thus a static manifesta-tion of the observed jamming transition.To study the IFJJA with anisotropic Josephson cou-plings, we consider a classical model described by thefollowing Hamiltonian with the Josephson coupling be-tween nearest neighbors along the x and y axes: H ( λ, { φ i } ) = − X i h cos φ xi + λ cos φ yi i (1)with the gauge-invariant phase differences across thejunctions, φ i = ( φ xi , φ yi ) ≡ ( θ i +ˆ x − θ i − A xi , θ i +ˆ y − θ i − A yi ) (2)Here θ i ( i = 1 , , . . . , N ) is the phase of superconductingorder parameter on the i -th island located at the latticepoint r i = ( x i , y i ) of a square lattice of size N = L × L with 0 ≤ x i < L , 0 ≤ y i < L , r i +ˆ x = ( x i + 1 , y i ) and r i +ˆ y = ( x i , y i +1). The parameter λ denotes the strengthof the coupling anisotropy. Due to the symmetry forpermutation of axes x and y , it is sufficient to investigateonly λ ≥
1. The presence of the external magnetic fieldis described by the vector potential A i = ( A xi , A yi ). It isrelated to the filling factor f , which is the average numberof flux quanta per square plaquette, by taking its latticerotation rot A i = 2 πf with rot X i ≡ X xi + X yi +ˆ x − X xi +ˆ y − X yi .Specifically we employ a commonly used irrationalnumber (3 − √ / ≈ .
382 for f [10]. We believe, how-ever, that the properties we discuss below do not de-pend on the specific choice of the irrational number. Toimpose a periodic boundary condition, f must be ap-proximated by a rational number. It is known that agood approximation is obtained by using the Fibonacciseries { F n : F n +2 = F n +1 + F n } as f = F n − /F n .We use two series F n = · · · , , , , , · · · and F ′ n = · · · , , , , , · · · corresponding to different initialconditions ( F = 1 , F = 2) and ( F ′ = 1 , F ′ = 3). Weuse system sizes L = F n to ensure truly irrational fillingin the thermodynamic limit L → ∞ . II. VARIATION ANALYSIS
We now study the GS of the system by extending theprevious analysis that was limited to the computation
FIG. 1. Vortex configurations of the metastable states (can-didates for the GSs) with various FRLV q = ( q x , f ) in theanisotropic IFJJA with λ ≥
1. White squares indicate theplaquettes containing vortices. The panels from left to rightcorrespond, respectively, to q x L =13,14,15,16, and 17 with L = 34. As the anisotropy λ increases, the GS changes fromthe left to right. Note that there is no recursive unit exceptthe L × of a few terms in the 1 /λ expansion [13]. First we per-formed numerical searches of the GSs at various strengthsof the anisotropy λ on small systems using a simulatedannealing method (not shown here). We found that thevortex configurations in the GSs exhibit stripe patternswhose tilt angle with respect to the axis of the weakercoupling varies with λ (see Fig. 1). This comes fromthe fact that φ i has one-dimensional periodicity, whichis generally incommensurate with the underlying JJA,and a fundamental reciprocal lattice vector (FRLV) q ( λ )varies with λ . We utilize this fact in the following anal-ysis. In the large anisotropy limit it was found to be q GS ( ∞ ) = (1 / , f ) (shown in the right most panel inFig. 1) [13].Let us now explain our strategy, which is slightly re-formulated but equivalent to the scheme presented inRef. [13] except for some partial use of numerical pro-cedures. First, recalling Eq. (2), we find that the phasedifferences must satisfy the following equation at eachplaquette: rot φ i = − rot A i = − πf. (3)To meet this condition, it is convenient to decompose φ into a uniform-rotation part and a rotation-free part as φ i = φ ∗ i + n max − X n =1 ϕ n e nπi q · r i . (4)where rot φ ∗ i = − πf rot (cid:2) ϕ n e nπi q · r i (cid:3) = 0 . (5)The upper bound of the harmonics n max is given by thesmallest n with which both nq x and nq y are integers. Italmost always equals L in the present model and becomesinfinite in the irrational limit. The first of Eqs. (5) can besolved by φ ∗ i = ( − πf y i + C q , C q is a constantintroduced to satisfy the periodic boundary condition for θ i . The second of Eqs. (5) can be solved by ϕ n = ( ϕ xn , ϕ yn )with ϕ yn /ϕ xn = (1 − e nπiq y ) / (1 − e nπiq x ), which can beused to eliminate the degrees of freedom { ϕ yn } in favor of { ϕ xn } (or vice versa).We assume that the FRLV can be parametrized as q = ( q x , f ) by a parameter q x , which is suggested bythe 1 /λ expansion analysis [13]. This perturbative anal-ysis suggests that a unique stable solution, which satisfiesthe current conservation (force balance) condition, can beconstructed explicitly for a given q x that is left as a con-trol parameter. The analytic computation is, however,quite cumbersome and difficult to continue to higher or-ders. To overcome the difficulty, we numerically searchthe configuration with minimal energy, E m ( λ, q ) = min { ϕ n } H ( λ, q , { ϕ n } ) , (6)for a given FRLV. We performed this analysis in therange f ≤ q x ≤ / { ϕ xn } .Actually, we numerically integrate the equations of mo-tion; dϕ xn /dt = − ∂H/∂ϕ xn for 1 ≤ n < n max , with aninitial condition ϕ xn = 0 for all n . We determine thatthe system relaxes to the minimum-energy state when | dH/dt | becomes less than 10 − . As an example we showin Fig. 1 the vortex configurations corresponding to thecandidate FRLVs for L = 34.The final step is to determine q GS ( λ ): q which givesthe minimal energy for given anisotropy λ as E GS ( λ ) = min q E m ( λ, q ) = E m ( λ, q GS ( λ )) . (7)Although there is no rigorous proof that the states ob-tained by this method are the true ground states, we findthat they coincide with the ones obtained by the simu-lated annealing methods. We thus believe that they arethe true ground states. III. RESULTSA. successive structure transition
Figure 2(a) shows the minimal energy E m ( λ, q ) as afunction of q x for λ =1.0, 1.5, and 2.0. From this wedetermine q GS and E GS for a given anisotropy λ . Theenergy behaves quadratically in the vicinity of each mini-mum as E m ( λ, q ) − E GS ( λ ) = L c ( λ )[ q − q GS ( λ )] , where c ( λ ) is a constant of O ( L ). Thus there is a continu-ous band of metastable states with slightly different FR-LVs around the GS. (We confirmed that the obtainedmetastable states are stable even if we lift the constraint;the Fourier components equal zero except for the har-monics n q .)Figure 2(b) shows the λ dependence of the x compo-nent of the q GS ( λ ) = ( q x GS , f ) with α = x, y . It can beseen that q x GS monotonically increases with λ and changesin a stepwise manner at several points, where level cross-ings occur between metastable states with neighboringFRLVs. As L becomes larger, the number of steps in-creases and q x GS tends to be a continuous function of λ .It can be seen that the horizontal stripe state q = (1 / , f )[13] is no longer the GS for λ ≤ .
8. The vortex stripes E m / N + ( ) q x L 18 34 76 144 322 (a)
L 8 18 34 76 144 322 q x G S (b) FIG. 2. (Color online) (a) Minimal energy vs q x for variousvalues of λ . Constant values are added in the cases of λ = 1 . . q GS . in the GS become more tilted with respect to the weakercoupling axis when approaching the isotropic point λ = 1(see Fig. 1).The GS of the isotropic system ( λ = 1) is of particularinterest. For small sizes L ≤
18, the GS is the previ-ously found staircase state [17–19] with q x GS = q y GS = f .However, this is not the case for L ≥
34. A new min-imum appears at q x ≈ .
416 and there is a local min-imum at q x ≈ . f = 0 . · · · . We obtain q x GS =14/34, 23/55, 37/89,60/144, and 97/233, where the numerators constitute aFibonacci series (with F ′′ = 1 and F ′′ = 4) and thedenominators correspond to L . Therefore, q x GS seems toequal 0 . · · · [ > (3 −√ /
2] in the limit of L → ∞ .A notable feature is that the symmetry for the permuta-tion of x and y axes spontaneously breaks in contrast tothe symmetric staircase state mentioned above. (a) ~ x /2 z ~ y /2 z(b) FIG. 3. (Color online) Hull function of (a) φ x and (b) φ y forsix different values of λ . Each graph has L (=144) points. Thearrows indicate the direction of decreasing λ . B. breaking of analyticity in hull functions
Next we investigate more closely the properties of theGSs parametrized by the FRLVs q , which are generallyincommensurate with respect to the underlying JJA. Auseful measure of such an incommensurate object is theso called hull function [14, 16, 20] ˜ φ ( z ) = ( ˜ φ x ( z ) , ˜ φ y ( z )),with which the phase field is written as φ i = ˜ φ ( q · r i ) . (8)In Fig. 3 the hull functions ˜ φ x ( z ) and ˜ φ y ( z ) are plottedwith respect to 0 < z <
1. It is noteworthy that ˜ φ x isdistributed, covering all phase from − π to π , while ˜ φ y is bounded around zero to have a gap around ˜ φ x = ± π .When the hull function is smooth and gapless, there is asliding soft mode [13]; the infinitesimal increment of φ x is compensated for by the shift in the argument of thehull function q · r i , which costs no energy barrier. Sincethis leads to dissipative vortex flow driven by currentinduction, the GSs have superconductivity only along thestronger-coupling direction. L = 144 ~ d y / d z z(a) ~ = 1 L 34 76 144 322 d y / d z z(b) FIG. 4. (Color online) Derivative of the hull function of φ y .(a) Anisotropy λ dependence for L = 144. (b) Size L depen-dence at the symmetric point λ = 1. A remarkable feature is that the hull function in boththe x and y directions becomes distorted by decreasingthe anisotropy λ . While the hull functions are smoothwith larger anisotropies [13], steplike structures appearwhen approaching the symmetric point λ = 1. Thehull function at λ = 1 appears to be nonanalytic: Itis discontinuous at several points. Figures 4(a) and4(b) show the discrete derivative of the hull functions d ˜ φ α ( z ) /dz ≡ [ ˜ φ α ( z + 2 π/L ) − ˜ φ α ( z )] / (2 π/L ). It can beseen that sharp peaks emerge when approaching λ = 1.This is reminiscent of the critical behavior of the Aubrytransition in the Frenkel-Kontorova model [14, 16, 20].Figure 4(b) shows the size dependence of the derivativeat λ = 1. As the size L increases, not only do the peaksbecome sharper, but also the number of peaks increasesreflecting higher harmonics. We speculate that the hullfunction becomes discontinuous everywhere in the ther-modynamic limit. C. Scaling behavior and diverging characteristiclength
Let us examine further the singular behavior aroundthe symmetric point λ = 1 . Figure 5(a) shows the ampli-tude of the n th harmonic n q [note that q is set equal to q GS ( λ = 1) independently of λ ]. At λ = 1, | ϕ yn | seemsto decay as a power function of n with exponent close to − n ∗ ( λ ) for λ >
1. This cutoff indicatesthe smoothing of the hull functions. We found a scalingbehavior | ϕ x,yn ( λ ) || ϕ x,yn (1) | = F x,y ± (cid:18) nn ∗ ( λ ) (cid:19) with n ∗ ( λ ) ∝ | λ − | − ν (9)not only for λ >
1, but also for λ <
1, as shown inFig. 5(b). The exponent ν is roughly estimated as ν ≃ .
4. We confirmed that the same scaling works well forthe FK model with ν = 0 . λ = 1.We note that this observed singularity is present not onlyfor q GS (1), but also for other q ’s around q GS (1), i.e., forthe continuous band of metastable states.Initially, it would appear as if the characteristic lengthin Eq. (9) were 1 /n ∗ | q | and it went to zero, which is con-trary to ordinary critical behavior. However, we specu-late, inversely, that there is a diverging length scale forthe following reasons. On lattice systems, the compo-nent of the wave vector, nq , should be treated in the firstBrillouin zone ( − / , /
2) as q FBZ n ≡ nq − ⌊ nq + 1 / ⌋ ,where ⌊· · · ⌋ is the floor function. For irrational q , { q FBZ n } behaves like a series of random numbers made by the lin-ear congruential generator in a long span of n . Thus themaximum of 1 /q FBZ n for n ≤ n ∗ is roughly proportional to n ∗ . If q x and q y are independent irrational numbers, themaximum of the vector length is proportional to √ n ∗ .In addition, the number of harmonics in the finite-sizesystem is bounded by L when q is approximated by anirreducible fraction p/L (note that q FBZ n + L = q FBZ n ). Thisalso indicates that n is a measure of the length.The breakdown of the analyticity of the hull functionmeans that the sliding soft mode disappears; jammingoccurs even in the direction where the hull function isgapless. Indeed, of the transport properties of the presentsystem it has been found that the current-voltage charac-teristics exhibit a scaling feature around the symmetricpoint λ = 1 [12]. Thus Eq. (9) is regarded as a staticsignature of the jamming transition. IV. CONCLUSION
To summarize, we studied GSs and low lyingmetastable states of the IFJJA with anisotropic Joseph-son couplings. We found the GS changes continuously -6 -4 -2 smaller | -1| = 0.0, 0.08, 0.16, 0.32, 0.64 | y n | n slope -1L=377 (a) -2 -1 -4 -3 -2 -1 < 1| -1| = 0.06, 0.08, 0.12, 0.16, 0.24, 0.32, 0.48, 0.64 (b) L 199 233 322 377 | y n () | / | y n ( ) | n ( -1) > 1 FIG. 5. (Color online) Spectrum of the harmonics of the hullfunctions (a) before and (b) after scaling. We set ν = 2 .
4. Inthe scaling, we do not use the data for n > L/
4, which showa strong finite-size effect. with the variation of the anisotropy λ of the Joseph-son coupling between the horizontal vortex stripes in thestrong anisotropy limit and nearly (but not exactly) di-agonal vortex stripes at the isotropic point λ = 1.It is interesting to note again that the present systemadmits a continuous band of metastable states aroundthe GS, which is presumably the source of the glassinessin the present system. The analyticity breaking of thehull function approaching the isotropic point λ → ACKNOWLEDGMENTS
This work was supported by a Grant-in-Aid for Scien-tific Research (C) (Grant No. 21540386), a Grant-in-Aidfor Scientific Research on Priority Areas “Novel States ofMatter Induced by Frustration”(Grant No. 1905200*), and King Abdullah University of Science and Technol-ogy Global Research Partnership (Grant No. KUK-I1-005-04). [1] G. Tarjus, S. A. Kivelson, Z. Nussinov, and P. Viot, J.Phys.: Condensed matter , 1143 (2005).[2] J. F. Sadoc and R. Mosseri, eds., Geometrical frustration (Cambridge University Press, 1999).[3] M. Tinkham, ed.,
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