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Quantum and Thermal Depinning of a String from a Linear Defect

The problem of a massive elastic string depinning from a linear defect under the action of a small driving force is considered. To exponential accuracy the decay rate is calculated with the help of the instanton method; then, fluctuations of the quasiclassical solution are taken into account to determine the preexponential factor. The decay rate exhibits a kind of first order transition from quantum tunneling to thermal activation with vanishing crossover region. The model may be applied to describe nucleation in 2-dimensional first order quantum phase transitions.


I. INTRODUCTION
Macroscopic tunneling phenomena always attracted theorists' attention since it is the field where classical and quantum physics meet each other. It governs the low temperature decay of metastable states of physical systems with many degrees of freedom, separated from the neighboring states by high potential barriers; at high temperatures the metastable state decays via thermal activation. Several such systems were studied, including motion of dislocations across Peierls barriers [1,2], tunneling with strong friction [3], or quantum breaking of an elastic string [4].
In the present paper the problem of quantum and thermal depinning of a massive elastic string trapped in a linear defect and subject to a small driving force is considered. The interest in one-dimensional manifold tunneling has revived last years as it describes the creep of vortices trapped by columnar defects or plane twin boundaries [5]. Unlike thermal activation, which for nondisspative Hamiltonian systems is independent of the dynamic properties of the string [6], quantum tunneling is strongly dependent on the dynamics, which for the case of vortices in HTSC can be either dissipative or governed by a Hall force [5]. The tunneling problem of a massive string therefore cannot be directly applied to the vortex creep but its study is important since it allows for an exact analytical solution. Massive string depinning from a linear defect describes the nucleation of a new phase in the vicinity of a first order quantum phase transition in 2 dimensions. The boundary of the two phases may be considered as a string, the difference in chemical potentials of the phases plays the role of a driving force.
The problem is solved quasiclassically and exact solutions are obtained for the whole temperature range in the limit of small driving force. Then, fluctuations of the quasiclassical solutions are taken into account to determine the preexponentials. The special feature of the problem considered is the presence of a "first order" transition between the quantum tunneling and thermal activation regimes that should be contrasted with a smooth crossover between them in other related problems [1,2,7].

II. MODEL AND SUMMARY
Consider a massive elastic string on a plane pinned by a linear defect. The dynamics of the string is governed by the Lagrangian Here ρ is the mass density of the string, c is the speed of sound, and u is the displacement perpendicular to the string. The string is directed along the x-axis. In the presence of a driving force, V (u) can be represented as where V 0 (u) is a potential of a single pinning line of size u 0 : V 0 (0) = 0 and V 0 (|u| ≫ u 0 ) = E 0 . The driving force F renders the states localized near u = 0 metastable and the string depins and leaves the defect in a finite time Γ −1 . The lifetime Γ −1 will be obtained analytically for the whole temperature range in the limit of small F .
At zero temperature the string has no energy to overcome the potential barrier that separates the states with u = 0 and with u ≥ E 0 /F where the barrier disappears. In this case the decay rate is determined by macroscopic quantum tunneling. To study the tunneling process one has to rewrite the action in imaginary time (for temperature T ): and find the saddle point solutionũ(x, τ ) for this action that can be obtained from variation of (3). Such a solution of a Euclidean field theory is commonly referred to as an instanton [8]. Then, to exponential accuracy the lifetime The action can be easily estimated as follows [9,4]. Let τ be the tunneling time. Then the mass involved in the tunneling process can be estimated as M ∼ ρcτ . We can describe the collective tunneling process as a tunneling of one particle of mass M through a potential barrier of height E 0 cτ . The tunneling time can be obtained from the condition that the kinetic energy M (E 0 /F ) 2 /τ 2 should be of the order of the potential energy E 0 cτ . This gives the estimate τ 2 ∼ ρE 0 /F 2 and the action Such a quasiclassical treatment is valid provided that A Q /h ≫ 1. This criterion can be expressed as where α = F/F 0 and F 0 ∼ E 0 /u 0 is the typical value of the force that keeps the string in the well. F 0 is also of the order of the critical driving force F c at which the potential barrier disappears. The following consideration will be constrained to the limit of small α ≪ 1. At high temperatures when the periodicityh/T along the τ -axis becomes smaller than the size ∼ (ρE 0 ) 1/2 /F of the quantum instanton, the saddle point solutionũ(x) does not depend on τ any more and the escape rate is determined by pure thermal activation. In this case Γ −1 ∝ exp(−∆/T ), where ∆ is the energy of the optimal configurationũ(x) that extremizes (3). Note that it is independent of the dynamics since the thermal saddle point solution does not depend on imaginary time. The energy barrier ∆ can be estimated in a way similar to the one used for the quantum case. Dimensional analysis yields The activation energy diverges at zero driving force. Such a behavior should be contrasted with the thermal kinkantikink nucleation in the models with several degenerate minima (Sine Gordon, φ 4 theories) [1,2], where in the absence of a driving force the activation barrier remains finite and is twice as large as the kink rest energy E K . In the present case, the single minimum of V 0 (u) is nondegenerate, resulting in large F -dependent size of the instanton and divergent ∆.
There exist a temperature T c when the probabilities of quantum and thermal instanton nucleations become equal, Usually, this temperature indicates a crossover from quantum tunneling to thermal activation. This is the case for single particle tunneling [7], tunnelling of a string between two degenerate minima when a small driving force lifting the degeneracy is applied [1,2], and quantum breaking of an elastic string [4]. In the present problem the situation is different. It will be shown that for T < T c the string escapes due to quantum tunneling, temperature corrections to the action being exponentially small for α ≪ 1. For T > T c the lifetime is determined by pure thermal activation. Hence, at T = T c the system exhibits a kind of first-order phase transition from the quantum to the thermal regime and no crossover region in the conventional sense is obtained. Such a behavior is a consequence of the coexistence of two qualitatively different quantum and thermal instantons in the vicinity of T c . The total Γ −1 = (Γ Q + Γ T ) −1 is equal to Γ −1 Q for T < T c and to Γ −1 T for T > T c . On the other hand, in a usual situation there exist only one saddle point solution of (3) that describes quantum tunneling at low temperatures, pure activation at high temperatures, and crosses over between them at T ∼ T c .

III. QUASICLASSICAL ANALYSIS
In this section the saddle point solutions for the action (3) are obtained in the limit of small driving force. For further use I want to specify some quantities defined above by order of magnitude estimates. We will see below that in the limit of small F the quasiclassical results for the nucleation rate are independent of the certain features of V 0 (u) at the scale u 0 . The only relevant parameter is the depth E 0 of the potential. The other characteristic of V 0 (u) that determines instantons interaction and the strength of quantum fluctuations is the curvature at the origin V ′′ 0 (0). Instead of it I define the width of the pinning potential according to The ratio of the driving force to the pinning force is the small parameter in the problem: In the following I will work in the units ρ = c = 1. One has to rescale x → (ρc 2 ) −1/2 x, τ → ρ −1/2 τ to return to dimensional units. The classical equation of motion obtained from the action (3) is The function u(x, τ ) is periodic in τ with the periodh/T and vanishes at |x| → ∞. Eq. (10) defines a 2D nonlinear electrostatics problem, the charge density being dependent on the electrical potential u.
The solution of Eq. (10) behaves essentially different for u ≪ u 0 and for u ≫ u 0 (see Eq. (2) for the definition of V (u)). Let us define a curve Γ on the (x, τ ) plane where u = u 0 . It separates the plane into inner and outer domains B andB. In the outer domainB, u < u 0 that corresponds to the string lying in the pinning well V 0 (u). Inside the domain B, u > u 0 and the string is under the barrier.
The key point in the determination of u(x, τ ) from the nonlinear Eq. (10) is to obtain the solution in the outer domainB and to extract the boundary condition for the internal region of the instanton. The size of the instanton (R for the quantum instanton (16), or L for the thermal instanton (18); see Fig. 1 for geometry clarifying) is of the order of E 1/2 0 /F as estimated above. In the external regionB of the instanton, u rapidly falls off to zero at a distance a that can be estimated from Eq.
Since a is much smaller than the boundary curvature radius (R for the quantum instanton, or ∞ for the thermal instanton), one can neglect the curvature in the Laplace operator in a layer of width a along the boundary Γ of the instanton and write simply ∇ 2 ≃ ∂ 2 ∂n 2 , where n is the outward normal to the boundary of the instanton. Then Eq. (10) becomes the equation of motion of a classical particle in an inverted potential −V (u): The energy E = 1 2 du dn is an integral of the motion. Its value should be obtained from the condition at infinity, that gives E = 0. Applied to u = u 0 , the equation E = 0 gives the boundary condition for the interior problem: The other condition is u| Γ = u 0 . Thus, we arrive at the following problem: Eqs. (13) and (14) define a Dirichlet problem with a unique solution for a given contour Γ. The solution of Eq. (13) with the boundary condition (15) (von Neumann problem) is unique as well. The problem (13)-(15) is therefore strongly overdetermined and cannot be solved for a general contour Γ.
There are only two regions for which the solution of Eq. (13) satisfies both boundary conditions simultaneously. They are (i) the circle and (ii) the strip (see Fig. 1).
(i) Circle of radius R. The solution is where r 2 = x 2 + τ 2 . The radius can be obtained from (12): This is the quantum instanton.
(ii) Strip of width 2L parallel to the τ -axis. Then with This solution is time-independent and holds at large temperatures.
The solutions are shown schematically in Fig. 2.
To compute the action one has to know the solution of Eq. (10) everywhere. The contribution of the boundary layer to the action is not universal and depends on the features of the pinning potential V 0 (u) but is of order α when compared to the contribution from the interior region. Thus, to leading order in α, the action for the circle is (here and below dimensional units are restored) whereas for the strip (temperature T ) the result reads with ∆ 0 the energy cost of the optimal tongue (18), Quantum and thermal actions become equal at The strip (18) is a solution for all T . It will be shown in the next section that this solution becomes unstable below some temperature T 0 = 0.9T c . The circle solution exists as long as its diameter 2R is less than the periodicityh/T in τ . The quantum solution then ceases to exist at The dependence of the actions on temperature for the two solutions is illustrated in Fig. 3(a). The lifetime Γ −1 is determined by the solution with the minimal action. To exponential accuracy, This answer holds for all temperatures but not too close to T c . In a very narrow region around T c , where |1 − T /T c | < T c /∆ ∼ α 2h (5)), both solutions contribute to Γ −1 according to the formula Γ −1 = (Γ Q + Γ T ) −1 . Note that the last equation is not a bridging formula but an exact expression up to exponentially small corrections. This is a consequence of the coexistence of two types of instantons close to T c . It is also instructive to study the temperature dependence of the quantum tunneling probability. It appears to be very small due to the exponential decay of u at large distances. Actually, the distance between the boundary of the instanton (16) and its image is δτ =h/T −h/T 1 . Then according to Eq. (10) the overlap of the exponentially decaying tails of the instanton and its image is as small as u 0 exp[−(V ′′ (0)/ρ) 1/2 δτ ], resulting in In the vicinity of T 1 ,

IV. FLUCTUATIONS
Let us now take fluctuations near the classical instantons into account and determine the preexponential factors in (25). The starting point is the expression for the lifetime Γ −1 [10,3] where Z is the partition function The neighborhood of the saddle point solutionũ(x, τ ) of the equation δA[u]/δu = 0 gives the main contribution to the imaginary part of Z. I expand u nearũ according to where u α (x, τ ) are the eigenfunctions of the operator δ 2 A[u]/δu 2 : The lowest eigenvalue λ 0 is negative, resulting in the imaginary contribution to the partition function. λ 1 = 0 is also an eigenvalue of (31) related to the zero mode with respect to translation. For the thermal instanton (18), λ 1 = 0 is nondegenerate and u 1 (x) ∝ ∂ xũ corresponds to the zero mode along x-axis, while for the quantum instanton (16), the zero level is twofold degenerate since the location of the instanton on the x, τ plane depends on both coordinates.
Applying standard methods [11,12,3] for zero modes and the negative eigenvalues we arrive at the following expressions: The lifetime per unit length for thermal activation is The decay rate for quantum tunneling reads Here det ′ and det ′′ denote the determinants with the zero eigenvalues omitted.

A. High-temperature instanton
For the high temperature τ -independent solution (18) (I use the unitsh = ρ = c = 1), The eigenvalues of the operator (34) are given by where n is an integer and {ǫ m } is the spectrum of the one dimensional Schrödinger equation The potential U (x) is illustrated schematically in Fig. 4. L ∝ α −1 is the semiwidth of the instanton strip (18), the width of the minimum of U (x) near |x| = L, a ∼ Lα, is independent of α; U (|x| ≫ L) = ω 2 ≡ V ′′ 0 (0) is the oscillator frequency for the pinning potential V 0 (u). The form of the potential near |x| = L depends on the detailed features of V 0 (u). For α → 0, L → ∞ and the potential well U (x) becomes deep in the sense that as many as 4 π α −1 discrete levels exist in this potential. Excited states can be obtained within the quasiclassical approximation. It is possible also to obtain the low lying levels in the limit of small α.
Let us define γ as a function of the continuous parameter ǫ as where ψ(x) is the solution of Eq. (37) with energy ǫ that vanishes at x → ∞. Once the function γ(ǫ) is known, one can obtain the discrete spectrum in the following way. For |x| < L, the solutions of Eq. (37) are given by ψ ǫ (x) = cos( √ ǫx) and ψ ǫ (x) = sin( √ ǫx) for even and odd states respectively. The spectrum then can be obtained from the continuity of the logarithmic derivative of ψ ǫ (x) at x = L: The function γ(ǫ) is not universal and depends on the shape of the minimum of U (x), that is on the details of V 0 (u). It appears however that γ(0) is independent of the pinning potential. We may find γ(0) by using our knowledge that the first excited level has zero energy. For |x| < L its wave function linearly depends on x yielding The low lying states can be obtained from Eq. (39) with the right hand side set equal to γ(0) thus neglecting the difference between γ(ǫ) and γ(0) for small ǫ. Then one immediately obtains the implicit equations for the spectrum: for even states and √ ǫL cot( for odd states. The approximation γ(ǫ) ≃ γ(0) is valid for small enough energies ǫ ≪ ǫ * ∼ γ(0)/γ ′ (0). One can estimate γ ′ (0) as γ(ω 2 )/ω 2 ∼ 1/ω, yielding ǫ * ∼ α −1 /L 2 . On the other hand, the energy of the n-th excited level can be estimated as ǫ n ≃ (πn/(2L)) 2 for large n. Therefore, Eqs. (41) and (42) correctly describe as many as α −1/2 low lying levels.
The negative eigenvalue ǫ 0 can be obtained from Eq. (41) and is equal to where µ = 1.19968 is the solution of the equation µ tanh µ = 1.
The negative eigenvalue determines the temperature T 0 where λ 01 changes the sign leading to the instability of the thermal instanton (18) with respect to a small perturbation of its boundary. Inserting (43) into Eq. (36), I obtain in dimensional units, Now we are in a position to calculate the ratio of determinants in (32). To do it, one has to compute the products over n [2], resulting in with the quantum correction to the barrier height, Here δ(q) is the phase shift of the states in the continuous spectrum, ǫ(q) = ω 2 + q 2 , and M labels the last discrete state. The function χ is defined according to At large q, δ(q) can be calculated perturbatively [13], leading to a logarithmic divergence of the integral over continuous spectrum in (47) that should be cut off at some wave vector Λ: The ultraviolet cutoff Λ is a new parameter that should be introduced for the problem described by the Lagrangian (1) to be well defined. The value of Λ remains undefined in the model in question. Physically, it is determined by the length scale at which the elastic approximation of the bending energy in (1) breaks down. Then Λ may be estimated as the inverse core radius of the linear manifold considered. χ(T /T c ) is a complicated function of temperature. For T close to T 0 , only the lowest level with positive energy ǫ 2 is excited. Its energy can be obtained from Eq. (41): ǫ 2 = (2.79839/L) 2 , resulting in For intermediate temperatures, T 0 ≪ T ≪ T 0 /α ∼ ω, many levels contribute to the sum in (48). In this limit, ǫ m ∼ (πm/(2L)) 2 and For T ≫ T 0 , many levels are excited and the quasiclassical approach can be applied. The sum over the states in Eq. (48) can be expressed via an integral over phase space: The first term in (53) accounts for the two omitted states in Eq. (48). For T ≪ T 0 /α, Eq. (53) reduces to Eq. (52). For T ≫ T 0 /α, one finds Substituting (18) and (46) into (32), we obtain ∆ = ∆ 0 + ∆ 1 and B(T ) in conventional units are given by the following expressions:

B. Quantum instanton
Fluctuations of the quantum instanton (16) are governed by the following Hamiltonian where U (r) is cylindrically symmetrical, The potential U (r) looks very similar to the one-dimensional potential U (x ≥ 0) illustrated in Fig. 4. The ratio of determinants in (33) can be expressed in terms of the spectrum as where the first sum is taken over the states in the potential (59) and the second in the potential U (r) = ω 2 . We can estimate (60) within the quasiclassical approach by taking advantage of the large number of bound states in the potential (59) for small α. According to the quasiclassical quantization rule, each state occupies the volume (2π) 2 in phase space and (60) reduces to The integral over the momentum is logarithmically divergent, leading to Inserting (62) in (33), and using (16) rewritten in conventional units, I obtain the final result The results (56) and (64) are valid provided that the fluctuation contribution is less than the classical one. This provides the conditionh This is a much stronger condition than that of (5). When (65) fails then one has to take unharmonic fluctuations into account that will renormalize the numerical coefficient in front of (64), whereas the estimate (4) holds as long as the condition (5) is fulfilled.

V. CONCLUSION
The problem of a massive string depinning from a linear defect has been solved in the limit of small driving force (α ≪ 1) allowing for an exact analytical solution in the whole temperature range. It has been shown that there exists a temperature region T 0 < T < T 1 where the action (3) has two different saddle point solutions: (16) and (18). The actions for these solutions become equal at T = T c . For T < T c , the lifetime is determined by quantum tunneling (Eq. (63)) with exponentially small temperature corrections (27). For T > T c , the lifetime is given by a pure activation expression (55). At T = T c the system jumps from quantum to classical behavior. The dependence of ln Γ −1 vs. T at constant F is shown in Fig. 3(a). The possibility of such a first order transition between quantum and thermal behavior was first discovered in Ref. [14].
It is worth emphasizing that the first order transition is obtained for small α ≪ 1. As α increases, T 0 (α) and T 1 (α) come closer to each other and become equal at some α c ∼ 1, that is, there exists a tricritical point where the first order transition disappears. For α close but below α c , the temperature corrections to the quantum tunneling rate become large and the transition at T c (α) should be considered as a transition from thermally assisted quantum tunneling to pure activation.
Besides the transition at constant F , one can consider the transition at constant T by changing the applied force which may be easier to achive experimentally. ln Γ −1 as a function of F is shown in Fig. 3(b). For F < F c = 3π 2 1/2 T h (ρE 0 ) 1/2 , the metastable state decays via thermal activation with ln Γ −1 ∝ F −1 , whereas for F > F c , the string escapes via quantum tunneling with ln Γ −1 ∝ F −2 .
The problem in question is connected with the nucleation phenomenon in 2-dimensional first order quantum phase transitions. Consider a first order phase transition controlled by some parameter x. The quantum nature of the transition implies that it exists even at T = 0. One possible example is the liquid-solid transition in 2D helium films. Let x > 0 correspond to phase I, and x < 0 to phase II. The phase boundary is a string that has mass density ρ and elasticity ρc 2 = σ, where σ is the boundary line tension. The difference in chemical potentials of the phases is equivalent to the driving force F = µ I − µ II ∝ x. Suppose that phase II is prepared at x > 0. Such a state is metastable and is destroyed by the nucleation of phase I that can take place near the edges of the 2D system or in the bulk. Nucleation near the edges is exactly the problem of a massive string depinning from a linear defect. In the present case, the edge of the system plays the role of the pinning well with the depth E 0 = σ. It can be shown that nucleation always occurs near the edges since the bulk tunneling action and activation barrier are larger than the corresponding values for the edge: A bulk /A edge = 3π/2 5/2 and ∆ bulk /∆ edge = 16 √ 2/15. Finally, the results (56) and (64) for the string on a plane can be easily generalized to the string in (1+d) dimensions. The instantons (16) and (18) remain the same since any transverse displacement will cost extra bending energy. The preexponentials will be different; they will be multiplied by an additional factor from transverse fluctuations. The main effect is the appearance of a coefficient d in front of the logarithm in formulae (56) and (64).