Depinning of disordered bosonic chains
DDepinning of disordered bosonic chains
Nicolas Vogt,
1, 2
Jared H. Cole, and Alexander Shnirman
1, 3 Institut f¨ur Theorie der Kondensierten Materie,Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany Chemical and Quantum Physics, School of Applied Sciences, RMIT University, Melbourne, 3001, Australia L. D. Landau Institute for Theoretical Physics RAS, Kosygina street 2, 119334 Moscow, Russia (Dated: October 11, 2018)We consider one-dimensional bosonic chains with a repulsive boson-boson interaction that decaysexponentially on large length-scales. This model describes transport of Cooper-pairs in a Josepshonjunction array, or transport of magnetic flux quanta in quantum-phase-slip ladders, i.e. arrays ofsuperconducting wires in a ladder-configuration that allow for the coherent tunnelling of flux quanta.In the low-frequency, long wave-length regime these chains can be mapped to an effective model ofa one-dimensional elastic field in a disordered potential. The onset of transport in these systems,when biased by external voltage, is described by the standard depinning theory of elastic media indisordered pinning potentials. We numerically study the regimes that are of relevance for quantum-phase-slip ladders. These are (i) very short chains and (ii) the regime of weak disorder. For chainsshorter than the typical pinning length, i.e., the Larkin length, the chains reach a saturation regimewhere the depinning voltage does not depend on the decay length of the repulsive interaction. In theregime of weak disorder we find an emergent correlation length-scale that depends on the disorderstrength. For arrays shorter than this length the onset of transport is similar to the clean arrays,i.e., is due to the penetration of solitons into the array. We discuss the depinning scenarios forlonger arrays in this regime.
PACS numbers: 74.81.Fa, 74.50.+r, 73.23.HkKeywords: Depinning,Josephson junction array, Quantum-phase-slip ladder
I. INTRODUCTION
Depinning theory describes the onset of propaga-tion in many different physical systems. Examples in-clude electrical transport in charge density waves , thecritical current of magnetic flux lattices in type IIsuperconductors, the propagation of magnetic domainboundaries and crack formation in strained materials .It was recently shown that the onset of electrical trans-port in one-dimensional arrays of Josephson junctionsis also determined by the depinning of the charge-configuration along the array .In this paper we consider a more general model, a dis-crete chain occupied by Bosons with a repulsive inter-actions that decays exponentially on long length-scales.In such a model the interaction between neighbouringislands can be expressed by introducing a continuousvariable, quasi-charge/flux, whose value is determinedby the distribution of bosons along the chain. Assum-ing that the continuous variable changes adiabatically,an effective model can be derived with the help of theBorn-Oppenheimer approximation . In the case thatdisorder is present in the system, depinning theory canbe applied to find the critical driving force that leadsto a steady boson transport through the system. TheJosephson junction arrays studied in Ref. 7 represent aparticular realization of this model. Alternatively, ourresults describe the dual system of quantum phase slipladders. In the latter case (QPS ladders) the bosons aremagnetic flux quanta that tunnel through quantum phaseslip elements that separate the loops in a ladder. In voltage biased Josephson junction arrays, the depin-ning theory describes the transition from an insulatingregime at low voltages to a transport regime at highervoltages. The critical voltage of the transition is referredto as the switching voltage. The insulating regime of thearrays is governed by an effective sine-Gordon-like quasi-charge model.In the study of the onset of transport in Josephsonjunction arrays, Ref. 7, the connection to standard de-pinning theory was established under the assumption ofstrongly disordered background charges (also referred toas the maximal disorder model). Under this assumptionthe disorder-term in the effective model is spatially un-correlated, allowing one to apply the standard depinningtheory. Additionally the mapping to the standard de-pinning theory assumes the array length N to be muchlarger than all other length-scales of the problem.In this paper we study the depinning behavior of moregeneral chain models in a parameter regime that is notfound in Josephson junction arrays. Specifically we con-sider chains which do not meet one of the two aforemen-tioned conditions, e.g. the chains are either short or onlyweakly disordered. In the case of short chains we find asaturation regime. In the case of weak disorder, we findspatially correlated long-range disorder in the effectivesine-Gordon-like model. In the considered chain modelthe spatial correlation in the disorder decays approxi-mately exponentially for large distances. The depinningprocess in systems with spatial correlations that decaywith a power law has been studied in Ref. with thehelp of functional renormalization group theory. a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t II. THEORYA. The model
We consider a chain of islands, H = (cid:88) i,j
12 ˜ n i M i,j ˜ n j − (cid:88) i t (cid:16) b † i +1 b i + h.c. (cid:17) (1)˜ n i = N i − N − f i , (2)where N i are the discrete bosonic occupation numbers ofthe islands, N is the average occupation number at equi-librium (chemical potential, positive charge background), f i are the random gate charges, b i , b † i are the bosonic an-nihilation and creation operators ( N i = b † i b i ) and thebosonic tunnelling amplitude is given by t . In the limit N (cid:29) b i ≈ √ N e − iϕ i , b † i ≈ √ N e iϕ i ,which leads to H = (cid:88) i,j
12 ˜ n i M i,j ˜ n j − (cid:88) i E t cos( ϕ i − − ϕ i ) , (3)where E t ≡ N t/
2. For convenience we also introduce n i ≡ N i − N , so that ˜ n i = n i − f i .We assume that the long-range behavior of the inter-action matrix M i,j = M | i − j | is determined by an expo-nential decay on a length-scale Λ, M i,j ∝ e − | i − j | Λ for | i − j | ≥ Λ (4)One concrete example of this situation is a Joseph-son array, a chain of superconducting island coupled viaJosephson junctions and with self-capacitances C (ca-pacitances to the ground) and capacitances C betweenthe neighboring islands (junction capacitances). In thiscase E t = E J is the Josephson energy whereas the cou-pling matrix is given by M i,j = (2 e ) (cid:104) ˆ C − (cid:105) i,j , (5)where ˆ C = − C − C C + 2 C − C − C C + 2 C − C . . . . . . . . . . (6)This gives for the Fourier transform M ( k ) ≡ FT ( M i − j )and for M | i − j | the following expressions: M ( k ) = Λ E C + 2 (1 − cos( k )) , (7) M | i − j | ≈ Λ E C e − | i − j | Λ , (8)where the junction charging energy is defined by E C ≡ (2 e ) / C . In particular the energy cost of a single chargein such an array is of the order M j,j ≈ Λ E C . Activated behavior with activation energy of order Λ E C was ob-served in Ref. 13.It has been realized long ago that for Λ (cid:29) m i = i − (cid:88) j =1 n j , F i = i − (cid:88) j =1 f j . (9)The resulting Hamiltonian has the form, H = 12 (cid:88) i,j ( m i − F i ) U i,j ( m j − F j ) − (cid:88) i E t cos( θ i ) , (10)with the modified coupling matrix U i,j = U | i − j | = 2 M | i − j | − M | i − j |− − M | i − j | +1 . (11)Here θ i ≡ ϕ i − − ϕ i . One can easily check that m i and θ i are conjugate variables.A qualitative picture in the low energy, long wavelength regime can be obtained from the Fourier trans-form of the coupling matrice, U ( k ) = FT ( U i − j ) = 2(1 − cos( k )) M ( k ) , (12)where k ∈ [ − π, π ]. The k → M | i − j | is dom-inated by the exponential decay and for small k (cid:28) Λ − the Fourier transform of the interaction matrix M | i − j | isapproximately constant, M ( k ) ≈ M (0), which leads to U ( k ) ≈ M (0) k . (13) B. Standard Villain transformation
The model (10) can be treated by the standard tech-nique involving Villain approximation . We omit allthe details and only mention the fact that the spin wavepart of the resulting model in the limit k (cid:28) Λ − , where U ( k ) ≈ M (0) k is quadratic, corresponds to a Luttingerliquid , H = v (cid:90) d x (cid:20) K [ θ ( x )] + 1 K [ ∂ x m ( x )] (cid:21) , (14)[ θ ( x ) , m ( x )] = iδ ( x − x ) , (15)where the Luttinger liquid velocity v and the Luttingerliquid parameter K scale with the original model param-eters as v = (cid:112) M (0) E t , (16) K = (cid:115) E t M (0) . (17)The corrections to (15) due to vortices (phase slips) inthis type of theories (see, e.g., Ref. 17) are of the form ∝ cos(2 πp [ m ( x ) + F ( x )]) with p ∈ Z . The amplitude infront of this terms (fugacity of vortices) is predicted to besmall in the limit Λ (cid:29) so that without disorder thecritical value of K is close to 2 /π (it may be renormalizedby disorder ).We assume that M (0) (cid:29) E t and therefore K (cid:28) /π .In this case the system is firmly in the charge densitywave (CDW)-regime and dominated by classical chargedynamics. The disorder F ( x ) enters the relevant phaseslip terms ∝ cos(2 π [ m ( x )+ F ( x )]) and can pin the chargedensity profile. C. Alternative derivation
Here we generalize the derivation given in Ref. 8 forthe case of a Josephson array described by the capaci-tance matrix (6) to the case of a more general matrix M i,j characterized by a screening length Λ. We start byrewriting the Hamiltonian (10) as H = 12 (cid:88) j U ( m j − F j ) − (cid:88) j E t cos( θ j ) , + 12 (cid:88) i,j ( m i − F i ) δU i,j ( m j − F j ) , (18)where U = U j,j and δU i,j = U i,j − U δ i,j . Next wetransform the third term with the help of the Hubbard-Stratonovich transformation, which introduces a new de-gree of freedom Q i , often referred to as the quasi-charge.This gives H { Q } = 12 (cid:88) j U ( m j − F j ) − (cid:88) j E t cos( θ j ) , − U (cid:88) j Q j ( m j − F j ) − U (cid:88) i,j Q i (cid:2) δU − (cid:3) ij Q j , (19)such that H = min Q [ H { Q } ]. Next H { Q } = 12 (cid:88) j U ( m j − F j − Q j ) − (cid:88) j E t cos( θ j )+ 12 (cid:88) i,j Q i B i,j Q j , (20)where B i,j = − U (cid:2) δU − (cid:3) ij − U δ i,j . The Fourier trans-form reads B ( k ) = − U − U U ( k ) − U . (21)For small wave vectors k (cid:28) Λ − we obtain U ( k ) (cid:28) U and B ( k ) ≈ U ( k ) ≈ M (0) k . Thus, assuming Q i changesslowly enough as a function of the coordinate i , i.e., changes on length scales longer than Λ, we can approxi-mate H { Q } ≈ (cid:88) j U ( m j − F j − Q j ) − (cid:88) j E t cos( θ j )+ M (0)2 (cid:88) i ( Q i − Q i +1 ) . (22)For the Josephson arrays with the capacitance matrix(6) we obtain U = 2 E C and M (0) = 2Λ E C = 2 E C ,where E C ≡ (2 e ) / C = Λ E C . In this case the form ofthe third term of (22) is exact .The adiabatic dynamics of the model (22) without dis-order was analyzed in Ref. 8. The inclusion of disorder isstraightforward. The aim is to integrate out the degreesof freedom ( m i , θ i ). For a given (adiabatic) trajectory Q i ( t ) the dynamics factorizes to independent dynamicsof single junctions governed by the Hamiltonians H i ( Q i ) = 12 U ( m i − F i − Q i ) − E t cos( θ i ) . (23)The Born-Oppenheimer periodic potential is given bythe ground state of the well known Hamiltonian (23), E Q ( Q i + F i ), where Q i + F i serves here as the total quasi-charge. The function E Q ( Q ) is periodic with period 1 ascan be seen from (23). In the limit E t (cid:29) U ( E J (cid:29) E C )it is given by E Q ( Q ) = E S cos( Q ), where E S is the quan-tum phase slip amplitude .Thus we obtain the effective potential energy of thewhole array of the form U C = 12 (cid:88) i,j Q i B i,j Q j + (cid:88) j E Q ( Q i + F i ) ≈ (cid:88) i M (0)( Q i − Q i +1 ) + (cid:88) i E Q ( Q i + F i )= (cid:88) i E C ( Q i − Q i +1 ) + (cid:88) i E Q ( Q i + F i ) . (24)This potential is supplemented by the kinetic energy.In the limit E t (cid:29) U ( E J (cid:29) E C ) it reads T =(1 / (cid:80) i L ˙ Q i , where the L is the Josephson inductance L ≈ L J = 1 /E t = 1 /E J . The quadratic part of the La-grangian T − U C gives again the Luttinger liquid withthe parameters (16). Since we assume K (cid:28) /π , theperiodic potential E Q ( Q i + F i ) is relevant and pins thedensity profile. In what follows we investigate the chargepinning in this setup. D. Edge bias
From now on we employ the terminology of Josephsonjunction arrays and put 2 e = 1. As we are primarilyinterested in the transport properties of the chains, weintroduce a bias V at the edge: U C = (cid:88) i ( Q i − Q i +1 ) C + E Q ( Q i + F i ) − VC Q . (25)To simplify the treatment in terms of the depinning the-ory we transform the system from a boundary biased sit-uation to a spatially homogeneous driving by introducinga parabolic shift in Q and F ,˜ Q i = Q i − C C V ( N + 1 − i )( N − i )2 N , (26)˜ F i = F i + C C V ( N + 1 − i )( N − i )2 N , (27)and the corresponding potential part of the Hamiltonianwith a driving force
V /N , U C = (cid:88) i (cid:16) ˜ Q i − ˜ Q i +1 (cid:17) C + E Q (cid:16) ˜ Q i + ˜ F i (cid:17) + VN ˜ Q i C . (28)In this formulation the problem corresponds to the dis-crete version of the well known depinning problem in one-dimension . The elastic energy of the field Q i is deter-mined by the elastic constant C . The elastic field ispinned by the random pinning potential E Q ( ˜ Q i + ˜ F i )and driven by the homogeneous driving force V /N . Inthe pinned regime the applied force
V /N is not strongenough to overcome the potential barrier imposed on theelastic object Q i by the random pinning potential.In the case that no driving force is applied, V = 0, theform of the elastic object is determined by a competitionbetween the elastic term ( Q i − Q i +1 ) and the pinningterm E Q ( Q i + F i ) in U C . On small length-scales, wherethe elastic energy term dominates, Q i is approximatelyconstant. The field Q i changes on large length-scaleswhere the pinning potential dominates. The crossoverbetween the two regimes happens at the length scale L p ,which was first determined by Larkin for a flux line lat-tice in type II superconductors . The length L p goes bymany names depending on the physical systems that arepinned. In type II superconductors it is called Larkinlength, in ferromagnets with domain boundaries Imry-Ma length and for charge density waves it is calledFukuyama-Lee length . In this work we use the termLarkin length.Once the driving force V /N exceeds a critical force V cr /N , the pinning potential is overcome and the elasticobject starts to move through the disordered medium.An intuitive argument to find the value of the criticaldriving force can be found by comparing the driving forceto the pinning force at the Larkin length . The distri-bution of Q is rigid on length-scales up to the Larkinlength. The elastic object can only start to move whenthe driving force exceeds the collective pinning force ona segment with length L = L p . E. Strong disorder
We first consider the strongly disordered model forwhich the results of the standard depinning theory are directly applicable. To make the connection to these re-sults the difference between the original disorder ( f i ) andthe effective quasi-disorder before ( F i ) and after ( ˜ F i ) theparabolic shift in the quasicharge is important. In termsof the original disorder the strongly disordered model isdefined by, f i ∈ [ − / , / , (29) p ( f i ) = Θ H (cid:18) − | f i | (cid:19) , (30)where p ( f i ) is the probability distribution of the disor-der f i and Θ H is the Heaviside Θ-function. This modelcorresponds to the strongest possible disorder in the con-sidered chain model. A frustration f i with an absolutevalue larger than 1 / i -th island of the chain. Thedisorder is bounded by ± / f i itself is not spatially correlated, inthe effective quasi-charge model, the quasi-disorder ˜ F i iscorrelated between different islands i and j , (cid:68) ˜ F i ˜ F j (cid:69) dis (cid:54) = 0 for i (cid:54) = j . (31)At first this seems to deviate from the normal situation indepinning theory where the disorder in the system is notspatially correlated . However, in the depinning theory,only correlations in the pinning potential are relevantto the behaviour of the system. The potential E Q isa function of the quasi-charge with a periodicity of 1.Since the disorder f i is box distributed in an intervalthat corresponds to the periodicity of the potential, theoffset ˜ F i can be absorbed into another uncorrelated box-distributed disorder term f bi ,˜ F i = F i − + C C V ( N + 1 − i )( N − i )2 N + f i → f bi , (32) E Q (cid:16) ˜ Q i + ˜ F i (cid:17) → E Q (cid:16) ˜ Q i + f bi (cid:17) , (33) f bi ∈ (cid:20) − , (cid:21) , (34) p ( f bi ) = Θ H (cid:18) − (cid:12)(cid:12)(cid:12) ˜ f i (cid:12)(cid:12)(cid:12)(cid:19) . (35)From the point of view of the potential E Q , the quasi-disorder ˜ F i is equivalent to a spatially uncorrelated dis-order term f bi in the maximally disordered model.Another way to determine whether spatial correlationsin ˜ F i affect the quasi-charge model, is to calculate thedisorder-averaged correlation function of the pinning po-tential: (cid:68) E Q (cid:16) ˜ Q + ˜ F i (cid:17) E Q (cid:16) ˜ Q + ˜ F j (cid:17)(cid:69) dis = R ( Q − Q ) δ i,j , (36)where the correlation function R ( Q ) is given by, R ( Q ) = (cid:90) − dF E Q ( Q + F ) E Q ( F ) . (37)Since the correlator of the pinning potential is propor-tional to a Kronecker delta, the pinning potential is notspatially correlated.We have now seen that for the maximal disorder modelwe arrive at an effective model that conforms with thestandard assumptions of depinning theory. In this casethe Larkin length and the critical driving force are wellknown (see for example Ref. 1).The approximate value of the Larkin length in one-dimensional systems is given by , L p = 3 − Λ (cid:20) ˜ R (cid:18) E J E C (cid:19)(cid:21) − . (38)The relevant parameters of the chain are the energy E C ,the tunnelling amplitude E J , the chain length N and Λ.To express the Larkin length in terms of these parameterswe have defined the function ˜ R ,˜ R (cid:18) E J E C (cid:19) = (cid:32) (cid:0) E max Q (cid:1) E C (cid:33) , (39)where E max Q is the amplitude of the random pinning po-tential E Q ( ˜ Q i + ˜ F i ). The correlation function ˜ R is afunction of the dimensionless ratio of the tunnelling ma-trix element and E C . The function needs to be deter-mined numerically only once for all possible values ofchain length and C .Similarly the depinning force can be expressed in termsof ˜ R and is given by , V cr ≈ N C l L p (40)= N Λ − (cid:26) ˜ R (cid:18) E J E C (cid:19)(cid:27) . (41)Further corrections to this intuitive approach can be ob-tained from renormalization-group-approaches . Weuse the approximation Eq.40 in this work.In most systems, where depinning theory is applicable,the system size is much larger than the Larkin lengthand it is a good approximation to assume infinite systemsize. We now turn our attention specifically to short finitechains. From Eq. 41 we see that the critical driving forcedecreases with increasing Λ. At the same time the Larkinlength increases, L p ∝ Λ . (42)In finite chains the Larkin length becomes equal to thesystem size N when Λ reaches the value,Λ N = N (cid:26) ˜ R (cid:18) E J E C (cid:19)(cid:27) . (43)Increasing Λ further while keeping E C constant only in-creases the elastic energy E C coupling neighbouring is-lands. The Larkin length, the length-scale on which Q is approximately constant, should increase. However inthis limit the Larkin length is equal to the system sizeand therefore the field Q is approximately constant alongthe whole array.For Λ (cid:29) Λ N the critical driving force is independentof the interaction length as long as E C is kept constant.A lower boundary for V cr is approximately given by V cr ≈ √ N − (cid:26) ˜ R (cid:18) E J E C (cid:19)(cid:27) , (44)which is the critical driving V cr one finds for Λ = Λ N .In reality V cr saturates for smaller Λ, when N is of thesame order of magnitude as L p (for comparison see thenumerical simulations in Sec. III). This leaves the prin-cipal behaviour of Eq. 44 unchanged and contributes aprefactor of order one in the expression for the criticaldriving force. F. Weak disorder
In the weak disorder case the bare disorder f i is notevenly distributed in the interval [ − / , / . In such a system, su-perconducting wires are arranged in a ladder configura-tion, such that a one-dimensional chain of superconduct-ing loops is formed. The superconducting wires that areshared by neighbouring loops contain a very thin sectionthat forms the QPS-junction. Magnetic flux quanta inthe loops assume the role of the bosonic particles. TheQPS-junctions between the superconducting loops pro-vide the hopping matrix element and the coupling matrix M i,j is the inverse inductance matrix of the system. In aladder configuration of superconducting wires the induc-tance matrix has the exactly the previously mentionedtridiagonal form Eq. 6, as long as the kinetic inductancedominates over the geometric inductance.Due to the lack of large magnetic dipoles in the vicinityof such a system, a weak disorder limit is more likely tobe realized than in Josephson junction arrays.We consider two models of weak disorder: (i) the weakbox disorder f i ∈ (cid:104) − γ , γ (cid:105) , (45) p ( f i ) = 1 γ Θ H (cid:16) γ − | f i | (cid:17) , (46)with the disorder strength γ <
1; (ii) Gaussian disorder p ( f i ) = 1 σ √ π e − f iσ , (47)with a standard deviation σ < / F i +1 can not be neglected in the ar-gument of the effective potential E Q ( Q ). The maximalvalue of the disorder f i is smaller than the periodicity ofthe potential E Q and the long range correlation in thequasi-disorder ˜ F i can not be absorbed in the potential.The correlation function of the pinning potential there-fore acquires a long range correlation component. Wedecompose the correlation function into short and long-range components, (cid:68) E Q (cid:16) Q + ˜ F i (cid:17) E Q (cid:16) ˜ F j (cid:17)(cid:69) dis = R ( Q ) δ i,j + R ( Q, i, j ) , (48)with the δ -correlated component R ( Q ) and the longrange correlation function R ( Q, i, j ). Due to the longrange correlations the intuitive picture of the depinning-transition is not valid anymore. For a long range corre-lation function, R ( Q, i, j ) ∝ | i − j | − a , (49)that decays with a power law, the problem has beenapproached with the functional renormalization groupmethod (FRG) in Refs. 12 and 21. It has been shown that these long-range correla-tions lead to the emergence of a new length-scale in thepinned system, the typical correlation length L corr . Theroughness function w ( x ) of a pinned system shows a dif-ferent behaviour, namely a variation in the roughnessexponent ζ rough , depending on whether the system isprobed at length-scales smaller or larger than the cor-relation length . We derive typical correlation lengthsfor the two weak disorder models under the assumptionthat E Q can be approximated as a cosine-potential, E J ∼ E C , (50) E Q ( Q ) ≈ E maxQ [1 − cos (2 πQ )] . (51)To calculate the correlation function of the pinning-potential of two different chain sites j and k we set, with-out loss of generality, j < k . The correlation function inthe weak box-disorder model is given by an integral overthe disorder, R ( Q, j, k ) = (cid:0) E maxQ (cid:1) (cid:90) ∞−∞ d F j ˜ p ( F j ) (cid:18) γ (cid:19) k − j (cid:90) γ − γ d f j . . . (cid:90) γ − γ d f k − cos ( Y ) cos ( Y ) , (52) Y = Q + F j + V ( N + 1 − j )( N − j )2 N , Y = Q + F j + k − (cid:88) l = j f l + V ( N + 1 − k )( N − k )2 N , (53)where ˜ p ( F j ) is the probability distribution of the quasi-disorder F j . Expressing the cosine in terms of exponen-tials one finds that the absolute value of the correlationfunction R is bounded by an envelope function R E , | R ( Q, j, k ) | ≤ R E ( Q, k − j )= 2 (cid:0) E maxQ (cid:1) (cid:18) sin ( πγ ) πγ (cid:19) k − j . (54)The long-range correlation function decays exponentiallywith the distance k − j and the correlation of the pinning-potential decays on the length-scale, L corr = − (cid:16) sin( πγ ) πγ (cid:17) . (55)As expected the correlation length goes to zero in thelimit of the maximal disorder and diverges in the cleanlimit without disorder, γ → ⇒ L corr → , (56) γ → ⇒ L corr → ∞ . (57)For a Gaussian distribution of the bare disorder f i , the correlation function is bounded by the exponential func-tion R G , | R ( Q, j, k ) | ≤ R G ( Q, k − j ) = 2 (cid:0) E maxQ (cid:1) (cid:16) e − π σ (cid:17) k − j . (58)The correlation length is determined by the standard de-viation σ of the bare disorder, L corr = 12 π σ . (59)We can again test the limits of infinitely broad and non-disordered distributions, σ → ∞ ⇒ L corr → , (60) σ → ⇒ L corr → ∞ . (61)In the broad limit the Gaussian disorder shows the sameasymptotic behaviour as the box-disorder distributionwhen approaching the maximal disorder limit. The max-imal disorder limit is consistent with a very broad bareGaussian distribution. In the opposite limit the Gaussiandistribution corresponds to a homogeneous shift in thedefinition of the quasi-charge and the correlation lengthdiverges.The correlation length L corr marks the crossover be-tween a disorder free and a strongly disordered array.On length-scales smaller than the correlation length thevalue of the disorder F i is approximately constant andconstitutes a mere shift in the field Q . If the weaklydisordered system is probed on these length-scales it be-haves like a clean chain. On larger length-scales the valueof the disorder changes significantly and the system be-haves like a disordered chain. This transition is shown inthe next section with the example of the dependence ofthe threshold voltage on the length of the chains. III. SIMULATIONS
We obtain the critical driving force V cr by numericallysolving the equations of motion of the field Q i that canbe obtained from the Hamiltonian (28), M ¨ Q i + 2 Q i − Q i − − Q i +1 C + α R ˙ Q i + V Q ( Q i + F i ) = 0 , (62) M ¨ Q + Q − Q C + α R ˙ Q + V Q ( Q ) = VC , (63) M ¨ Q N +1 + Q N +1 − Q N C + α R ˙ Q N +1 + V Q ( Q N +1 + F N +1 ) = 0 . (64)The function V Q is the pinning force given by, V Q ( Q ) ≡ ∂ Q E Q ( Q ) . (65)To guarantee numerical convergence we have introduceda mass M and a linear dissipative term with a dissi-pation constant α R . Similar numerically simulations ofthe switching voltage in arrays of normal tunnel contactshave been conducted in Ref. 23.The critical driving force V cr is determined by adia-batically applying the boundary force V and determiningwhether a stable solution for the field Q i can be found.Although V is increased slowly, the switch-on time of thedriving force V in the numerical simulation is finite. Thephenomenological dissipative term has to be included tocompensate the small transport velocity ˙ Q i introducedby the switch-on of V . The introduction of a phenomeno-logical term is also a standard tool in the derivation of thedepinning force V cr in renormalization-group-treatmentsof pinned systems .The mass M and the dissipation parameter α R bothaffect the dynamical properties of the system, howeverthey have no influence on the breakdown of the staticstate. In the example of a Josephson junction array,the mass M corresponds to an inductance and α Q cor-responds to an Ohmic resistance. In a quantum phaseslip ladder M corresponds to a capacitance. In all simu-lations we choose the tunnelling amplitude and the cou-pling energy to be equal, E J = E C , so that the potential N V c r Λ = 2 Λ = 5 Λ = 7 Λ = 10 analytic estimateFIG. 1. (Color online) The critical driving force V cr of theclean chain is plotted as a function of the chain-length forseveral values of Λ. As long as the chain is more than twice aslong as Λ, V cr is independent of the length N and proportionalto Λ. The critical driving force has the value predicted by ananalytic estimate by Haviland and Delsing . In the regionwhere the chain is shorter than Λ the system is in the zero-dimensional limit. The critical driving force is proportionalto N and does not depend on Λ. E Q is close to a cosine potential. The length of the chain N and the parameter Λ are varied. A. The clean chain
We first simulate the clean model that has been used asthe default model in a number of experimental papers onJosephson junction arrays . While this model doesnot take into account charge disorder , it might be morerelevant for quantum-phase-slip-arrays than Josephsonjunction arrays as the former lack the strong charge dis-order that can be found in the latter.In the clean case, a simple argument to determine thecritical driving force V cr can be found in Ref. 24. In thecontinuum limit (Λ (cid:29)
1) for long chains (Λ (cid:28) N ) theeffective model of the clean chain is equal to the sine-Gordon model with a modified potential. The solutionsof the standard sine-Gordon equation of motion are thewell known solitons , Q ( x ) = 2 π arctan (cid:16) e γ sol x − vt Λ (cid:17) , (66) γ sol = 1 (cid:113) − v LC , (67)with the soliton velocity v . The spatial derivative of astatic soliton v = 0 has a maximal value of ∂ x Q ( x ) | v =0 ≤ π . (68) Λ V c r [ / C J ] N = 50 100 150 195 linear P-NFIG. 2. (Color online) The critical driving force V cr as afunction of Λ for different chain-lengths N in the clean chain.As long as Λ is larger than 2, the analytic estimate Eq. 70is reproduced and V cr ∝ Λ, as it is shown by the linear fit(dashed line) in the plot. For smaller Λ, non-propagating Q -excitations can be created in the chain by the adiabaticswitch-on of the driving force. The number of excitationsis proportional to the chain-length and the critical drivingis proportional to N -times the depinning-force of one exci-tation. The depinning force has been fitted (red lines) toan exponential function V sol = βe − γ Λ as it arises from thePeierls-Nabarro-Potential . The boundary driving force V takes the form of a bound-ary condition on the spatial derivative at x = 0, ∂ x Q ( x ) | x =0 = C C V . (69)This can be used to estimate the maximal boundary force V for which a static soliton can exist at the array ends, V cr = 4 √ π (cid:114) C C V maxQ ∝ Λ , (70) V maxQ = max Q ( ∂ Q E Q ( Q )) . (71)In the Josephson junction arrays this force correspondsto the switching voltage at which the array switches frominsulating to transport behaviour.The critical driving force does not depend on the arraylength and is proportional to the interaction length Λ.Both features are confirmed by the numerical simulationsin Fig. 1 and Fig. 2.In the limit Λ > N the spatial dependent field Q i takesthe same value on all islands of the chain, Q i → Q and thecoupled equations of motion simplify to a single equationof motion, M ¨ Q + α R ˙ Q + V Q ( Q ) = VC . (72)The one-dimensional clean chain model reduces to a zero-dimensional model. The critical driving force increaseslinearly with array size and is independent of Λ (Fig. 1). N V c r Λ = 2 analytic Λ = 2 Λ = 5 analytic Λ = 5FIG. 3. (Color online) The critical driving force V cr is plot-ted as a function of the length N of the disordered chain. ForΛ = 2 (black crosses) the critical force grows linearly with thechain-length N as expected from the analytic estimate (blackline). For Λ = 5 (blue triangles) V cr is proportional to N atlarger chain-lengths when N ≈ ≈ . L p and fits the ana-lytic estimate (blue solid line) from Eq. 41. Due to the strongdependence on the random disorder-configuration the lineardependence is only realised on average over 20 disorder con-figurations. The error-bars give the standard-deviation of thecritical driving force in the sample of disorder-configurations. When the interaction length Λ is comparable to theinter-site distance Λ < N and the Λ-dependence can be fitted to an exponentialbehaviour, V cr = N βe − γ Λ , (73)as seen in Fig. 2. Only one set of fitting parameters β, γ is used for all four simulated chain-lengths.The change of the switching voltage behaviour canbe understood in the following way. The interactionlength Λ is a measure for the ratio of the elastic cou-pling between neighbouring islands and the depth of thepinning-potential. For small interaction lengths Λ < Q -excitations can be created at the driven end of thechain without leading to complete depinning. Duringthe adiabatic increase of force V the whole chain isfilled with non-propagating Q -excitations. The depin-ning transition of these Q -excitations is determined bythe Peierls-Nabarro-potential . This give rise to aΛ-dependence of the form of Eq.73. In the context ofJosephson-junction-arrays this was discussed by Fedorovet al. for the depinning of a single 2 e -charge-excitation . B. The maximally disordered array
Here we present the critical driving force obtainedfrom numerical simulations of the maximally disordered V c r N = 50 100 150 195FIG. 4. (Color online) The critical driving force is plottedas a function of Λ in disordered arrays for a wide range of Λ.For Λ < L p is comparable tothe chain length and V cr is independent of Λ, see also Eq. 44.For intermediate Λ the behaviour (black rectangle) is shownin Fig. 5 2 3 4 5 6 7101520 Λ V c r N = 150 N = 195FIG. 5. (Color online) A comparison of V cr in the inter-mediate Λ regime of Fig. 4 (black rectangle) with a fittedpower-law decay (solid lines) and the analytic estimate Eq. 41(dashed lines) . From the fit we obtain an exponent of − . N = 150) and − .
56 ( N = 195) while depinning theory pre-dicts an exponent of − . model. In Fig. 3 we compare the dependence of V cr onthe parameter N with analytic estimate in Eq. 41. Atlarge N , where the array is longer than the Larkin length N > L p , we find that the numerical simulations fit tothe expected linear dependence on the system length.For small system lengths the switching voltage does notincrease linearly with N , as expected in the saturationregime where the Larkin length is comparable to the sys-tem size (Eq. 44).The numerically determined dependence of V cr on Λis shown in Fig. 4 and Fig. 5. For small Λ the inter- (a)0 50 100051015 N V c r γ = 1 γ = 0 . γ = 0(b)0 50 100051015 N V c r γ = 1 γ = 0 . γ = 0FIG. 6. (Color online) The critical driving force V cr is plot-ted as a function of the chain-length in a weak box-disordermodel. To enhance visibility we show two subplots for differ-ent disorder strengths: γ = 0 .
25, blue markers, subplot (a)and γ = 0 .
125 green markers subplot (b). For comparison the V cr of the clean case ( γ = 0, red asterisk) and the maximallydisordered model ( γ = 1, black crosses) are included in theplots. The behaviour of V cr changes when the chain length isequal to the correlation length N = L corr . Below N = L corr ,the critical driving force has approximately the same value asin the clean case. Above N = L corr it increases linearly with N as in the maximal disorder model, L corr ( γ = 0 . ≈ L corr ( γ = 0 . ≈ site distance is comparable to Λ and the continuum limitof the standard depinning-picture does not apply. Forlarge Λ the Larkin-length is comparable to the chain-length N and we observe a saturation of V cr with Λ. Thesaturation sets in for, N ≈ α sat L p , (74)where α sat is of order of one. Comparing the analyticestimate Eq. 44 with the saturation points we expect α sat in the range 2 . ≤ α sat ≤ . − (Eq. 41). Fitting the0numerical data to a power-law we obtain the exponents − . ± .
05 ( N = 150) and − . ± .
03 ( N = 195).However this is limited by the numerically accessiblechain-lengths and we can not obtain a robust confirma-tion of the value of the exponent of Λ from the numericalsimulations. C. Weak disorder and emergent correlation length
To validate our analytic model of the introduction ofa new length-scale by weak disorder, we also simulatethe depinning-transition of the weakly disordered chain.We choose the disorder strengths γ = 0 . L corr ( γ =0 . ≈
10 and γ = 0 . L corr ( γ = 0 . ≈
40. InFig. 6 it is shown that the system undergoes a transitionwhen the array-length is equal to the correlation length, N = L corr . Below N < L corr the chain is describedby the clean chain model ( γ = 0). Above the transitionthe critical driving force increases linearly with N . The N -dependence of V cr matches the maximally disorderedmodel γ = 1. When the correlation length is significantlylarger than the array size we can approximate all corre-lated disorder terms F i by a single value F i ≈ F . Theperfectly correlated disorder term F can be absorbed intothe definition of the quasi-charge and the system is equiv-alent to the clean array without disorder F i = 0.When the length of the chain exceeds the correlationlength one has to distinguish between two cases. Thecase when the correlation length is smaller than Λ andthe case where it is larger. The first case requires a carefultreatment to map the weakly disordered case to an effec-tive strongly disordered model. Here we limit ourselvesto the simpler second case. In this case one can under-stand the behaviour of the critical driving force with thefollowing simple argument.The typical length of a soliton Λ is smaller than the cor-relation length and static solitonic solutions of the field Q can exist in the chain. On the one hand this leadsto the creation of a boundary soliton at the edge of thedriven system that corresponds to the boundary solitonin the clean case. This gives rise to an offset criticaldriving force V offset cr . Since the chain is longer than L corr it can be subdivided into domains of length L corr . Toswitch into the conduction regime, the applied drivingforce needs to overcome the transport threshold in eachdomain, where the transport threshold is proportional tothe critical driving force in the clean chain Eq. 70, V cr − V offset cr ∝ (cid:114) C C V maxQ NL corr . (75)This mechanism explains the linear increase in V cr seenin Fig. 6 for N > L corr . IV. CONCLUSIONS
In this paper we have studied the depinning behaviourof discrete bosonic chain models that can be describedby an effective Hamiltonian in the adiabatic limit that issimilar to the sine-Gordon model. The most experimen-tally relevant realization of this model are linear arraysof Josephson junctions, however another possible realiza-tion is a ladder configuration of superconducting wireswith quantum phase slip elements separating neighbour-ing superconducting loops.We used analytical considerations and numerical sim-ulations to determine the critical driving force requiredto overcome the pinning of bosons in the chain. Inthe parameter regime that corresponds to experimen-tally studied arrays we reproduce the recently observedbehaviour . Going to new parameter regimes, namelyshort chains and weakly disordered chains, we see a sat-uration regime in short chains where the Larkin lengthexceeds the system length and the critical driving forceis independent of the decay length Λ of the repulsive in-teraction. In the weak disorder regime we observe theemergence of a new correlation length-scale L corr . Botheffects show good agreement between the analytic resultsand the numerical simulations ACKNOWLEDGMENTS
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