0.25

0.2

0.15

Fp

0.1

0.05

0

0 0.5 1 1.5

v

’2 1/2

Figure 12.3 Pinning force (in units of ν0 rp ) on a single vortex as a function

of velocity (in units of · ’1 rp ν0 ) obtained from the self-consistent theory. The

’2 1/2

curves correspond to the di¬erent temperatures T = 0.005, 0.05, 0.1, 0.2, 0.4, 0.5 (in

1/2

units of ν0 /rp ), where the uppermost curve corresponds to T = 0.005, and m =

3 ’1/2

0.1· 2 rp ν0 .

We ¬nd that the pinning force has a non-monotonic dependence as a function

of velocity, and that the peak in the pinning force decreases rapidly with increasing

temperature, and develops into a plateau once the thermal energy is of the order of the

average barrier height. At the highest temperature, the velocity dependence of the

478 12. Classical statistical dynamics

pinning force is seen in Figure 12.3 to approach the linear regime at low velocities

in accordance with the analytical result obtained in the high temperature limit,

Eq. (12.154). At high velocities, the pinning force is independent of the temperature

as apparent from Figure 12.3.

In fact, the pinning force is inversely proportional to the velocity at high veloci-

ties in agreement with the perturbation theory result, Eq. (12.142), as apparent from

Figure 12.4, where a comparison is made between the pinning force obtained from

lowest-order perturbation theory and the numerically evaluated self-consistent result.

The two results agree as expected in the large velocity regime, whereas the pertur-

bation theory result has an unphysical divergence at low velocities due to the neglect

of ¬‚uctuations, and a consequent absence of damping by the ¬‚uctuation exponent in

Eq. (12.152).

0.2

0.15

Fp

0.1

0.05

0

0 0.5 1 1.5

v

’21/2

Figure 12.4 Pinning force (in units of ν0 rp ) on a single vortex as a function of

velocity (in units of · ’1 rp ν0 ). The solid line represents the result obtained from

’2 1/2

the self-consistent theory, while the dashed line represents the result of lowest-order

3 ’1/2

1/2 ’1

perturbation theory in the disorder (T = 0.005ν0 rp and m = 0.1· 2 rp ν0 ).

In order to check the validity of the self-consistent theory beyond lowest-order

perturbation theory, we have performed numerical simulations. In Figure 12.5, a

comparison between the self-consistent theory and a numerical simulation of the

pinning force as a function of velocity is presented. The agreement between the

self-consistent theory and the simulation is good, except around the maximum value

of the pinning force, where the simulation is found to yield a higher pinning force

12.5. Single vortex 479

in comparison to the self-consistent theory. In this region the relative velocity ¬‚uc-

tuations are large, and in fact the self-consistent theory predicts that the relative

velocity ¬‚uctuations are diverging at zero velocity even at zero temperature, as we

discuss shortly. The self-consistent equations and their numerical solution, as well as

the simulations, can therefore be expected to be less accurate at low velocities.

0.16

0.14

0.12

Fp

0.1

0.08

0.06

0.04

0 0.5 1 1.5

v

’2 1/2

Figure 12.5 Comparison of the pinning force (in units of ν0 rp ) on a single vortex

as a function of velocity (in units of · ’1 rp ν0 ) obtained from the self-consistent

’2 1/2

’1 1/2

theory, solid line, and the numerical simulation, plus signs (T = 0.1ν0 rp and

’1/2

m = 0.1· 2 rp ν0

3

).

The convergence of the iterative procedure can be monitored by checking that

energy conservation is ful¬lled. The energy conservation relation is obtained by

multiplying the Langevin equation by the velocity of the vortex and averaging over

the thermal noise and the quenched disorder

m xt · xt = ’ xt · ∇V (xt ) + F·v + xt · ξ t .

+ · x2

™¨ ™t ™ ™ (12.157)

The ¬rst term is proportional to ‚t x2 , and vanishes since averaged quantities are

™t

independent of time, as the external force is assumed to be independent of time. The

¬rst term on the right-hand side, the term originating from the disorder, vanishes for

the same reason, since it can be rewritten as ’‚t V (xt ) . The energy conservation

™

relation therefore becomes, v = xt ,

· (xt ’ v)2 ’ xt · ξ t = ’v · Fp

™ ™ (12.158)

480 12. Classical statistical dynamics

or, in terms of the Green™s functions,

’i·‚t trGK = ’v · Fp .

2

+ 2·T ‚t trGR (12.159)

tt tt

t =t t =t

where tr denotes the trace with respect to the Cartesian indices. The energy conser-

vation relation simply states that, on average, the work performed by the external

and thermal noise forces is dissipated owing to friction.

In order to ascertain the convergence of the iteration process, employed when

solving the self-consistent equations, we test how accurately the iterated solution

satis¬es the energy conservation relation. In Figure 12.6 the velocity dependence

of the left- and right-hand sides of the energy conservation relation, Eq. (12.159),

is shown. After at the most twenty iterations, the energy conservation relation is

satis¬ed by the iterated solution to within an accuracy of 1%.

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

0 0.5 1 1.5

v

Figure 12.6 The values (in units of ν0 · ’1 rp ) of the expressions on the two sides of

’4

the energy conservation relation, Eq. (12.159), are shown as a function of the velocity

(in units of · ’1 rp ν0 ). The dashed line and the plus symbols correspond to the

’2 1/2

’1/2

’1

1/2

left- and right-hand side, respectively (T = 0.05ν0 rp and m = 0.1· 2 rp ν0

3

). The

energy conservation relation is ful¬lled to within an accuracy of 1%.

In Section 12.7 we shall consider dynamic melting of the vortex lattice, and it is

therefore of interest to check the validity of the ¬‚uctuations predicted by the self-

consistent theory against direct simulations of the Langevin equation. In order to

check the accuracy of the velocity ¬‚uctuations calculated within the self-consistent

12.5. Single vortex 481

theory, we have performed simulations of the velocity ¬‚uctuations. In Figure 12.7,

the velocity ¬‚uctuations obtained from the self-consistent theory are compared with

simulations.

1.05

1.04

1.03

1.02

1.01

1

0 0.5 1 1.5

v

Figure 12.7 Longitudinal and transverse velocity ¬‚uctuations (in units of · ’2 rp ν0 )

’4

as a function of the average velocity (in units of · ’1 rp ν0 ). The solid and dashed

’21/2

lines represent the results for the longitudinal (parallel to the external force), (xt ’

™

2 2

v) , and transverse, yt , velocity ¬‚uctuations obtained from the self-consistent

™

theory, respectively. The plus signs and crosses represent the simulation results for

1/2 ’1

the longitudinal and transverse velocity ¬‚uctuations, respectively (T = 0.1ν0 rp

’1/2

and m = 0.1· 2 rp ν0

3

). At low average velocities the ¬‚uctuations approach their

thermal value, T /m, which for the parameters and units in question equals 1. At

intermediate average velocities the longitudinal velocity ¬‚uctuations are larger than

the transverse, owing to the jerky motion of the particle along the preferred direction

of the external force, before reaching the same value at high average velocities where

the e¬ect of the disorder simply acts as an additional contribution to the temperature.

The agreement between the self-consistent theory and the numerical simulations

is seen to be good, indicating that ¬‚uctuations calculated from the self-consistent

theory are quantitatively correct. At low average velocities the velocity ¬‚uctuations

approach their thermal value T /m. The relative velocity ¬‚uctuations diverge at zero

velocity even at zero temperature. This can be inferred from the energy conservation

relation, Eq. (12.158), and the sub-linear velocity dependence of the pinning force at

low velocities, as for example is apparent from Figure 12.3. At intermediate average

482 12. Classical statistical dynamics

velocities, the velocity ¬‚uctuations in the direction parallel to the average velocity

(chosen along the x-axis), the longitudinal velocity ¬‚uctuations, (xt ’v)2 , are found

ˆ ™

to be larger than the ¬‚uctuations perpendicular to the average velocity, the transverse

™2

velocity ¬‚uctuations, yt . The reason behind this is that at not too high velocities,

where the force due to the disorder is strong compared with the friction force, the

motion of the particle is jerky since the particle slowly makes it to the disorder

potential tops, and subsequently is accelerated by the disorder potential. Since the

average motion of the particle is caused by the external driving force, the jerky

motion and the velocity ¬‚uctuations are largest in that preferred direction. At high

average velocity, the longitudinal and transverse velocity ¬‚uctuations saturate and

are seen to become equal, owing to the strong friction force causing a steadier motion.

In this connection we should also mention that we have noticed from our numerical

calculations that the second term on the left-hand side of Eq. (12.158) is independent

of the average velocity (and disorder), as is also apparent by comparing Figures

12.6 and 12.7. This thermal ¬‚uctuation contribution to the velocity ¬‚uctuations is

therefore given by its zero velocity value, and according to Eq. (12.158) is speci¬ed by

the equilibrium velocity ¬‚uctuations and therefore determined by equipartition. The

saturation value of the velocity ¬‚uctuations can therefore be determined from the

energy conservation relation, Eq. (12.158). For example, in the case of a small vortex

3√

· 2 rp / ν0 , we can use the high velocity expression for the pinning force,

mass, m

Eq. (12.142), and obtain the result that the saturation value equals T /m+ν0 /8πrp · 2 ,

4

a result in good agreement with Figure 12.7. At high average velocity, the velocity

¬‚uctuations saturate, and the e¬ect of the disorder simply acts as an additional

contribution to the temperature.

12.5.5 Hall force

In this section the e¬ect of a Hall force is considered, and the previous analysis of

the dynamics of a single vortex is extended to include the Hall force

m¨ t + · xt = ±xt — z ’ ∇V (xt ) + Ft + ξ t .

™ ™ ˆ (12.160)

x

We shall use the self-consistent theory to calculate the pinning force, the velocity

¬‚uctuations, and the Hall angle

FH ±

θ = arctan = arctan , (12.161)

v·F

ˆ ·e¬

which can be expressed in terms of the e¬ective friction coe¬cient.

Analytical results

The inverse of the free retarded Green™s function acquires, according to Eq. (12.160),

o¬-diagonal elements

’i±ω

mω 2 + i·ω

’1

DR (ω) = (12.162)

mω 2 + i·ω

i±ω

12.5. Single vortex 483

and the free retarded Green™s function is given by

1 mω + i· i±

R

Dω = . (12.163)

’i±

(ω +i0) ((mω +i·)2 ’±2 ) mω + i·

√ 2

In the high-velocity regime, v ν0 /(·rp ), where lowest-order perturbation theory

in the disorder is valid, we can neglect the self-energies in the self-consistent expres-

sion for the pinning force, Eq. (12.152), i.e. we can insert the free retarded Green™s

function and neglect the ¬‚uctuation exponent. Since the free retarded Green™s func-

tion is antisymmetric in the Cartesian indices, only the diagonal elements make a

contribution to the pinning force. The diagonal elements of the free retarded Green™s

Ryy

Rxx

function are identical, Dt0 = Dt0 , and given by

’· ± ±t ±t

e’·t/m

’ cos

Rxx

Dt0 = θ(t) 1+ sin (12.164)

·2 + ±2 · m m

3√

· 2 rp / ν0 ,

and we obtain for the pinning force, for vanishing mass, m

·ν0

Fp = ’ v. (12.165)

4π(· 2 + ±2 )rp v 2

4

We observe that the pinning force is suppressed by the Hall force in the high-velocity

√

ν0 (· 2 + ±2 )’1/2 rp , and the high-velocity regime therefore sets in at a

’2

limit, v

lower value in the presence of the Hall force.

√ √

ν0 /rp , and moderate velocities, v < · ν0 /((· 2 +

At high temperatures, T

±2 )rp ), the Hall force has the opposite e¬ect, i.e. it increases the pinning force, as a

2

calculation similar to the one leading to Eq. (12.154) shows that the pinning force is

3√

· 2 rp / ν0 ):

(m

ν0 (· 2 + ±2 )

Fp = ’ v. (12.166)

8π·T 2 rp

2

We have found by solving the self-consistent equations numerically at high temper-

√

ature, T = 10 ν0 /rp , that the pinning force is linear at low velocities and increases

with increasing Hall force. The deviation from the linear behavior in the presence of

the Hall force starts at a lower velocity value in accordance with the high-velocity

regime starting at a lower value in the presence of the Hall force.

Numerical results

For any given average velocity of the vortex, the pinning force can be calculated

from the self-consistent theory. We have numerically calculated the pinning force

for various strengths of the Hall force. In Figure 12.8, the resulting pinning force

as a function of the velocity is shown for di¬erent strengths of the Hall force for a

√

temperature lower than the average barrier height, T < ν0 /rp . The Hall force is

seen to reduce the pinning force in this temperature regime except, of course, at low

velocities.

484 12. Classical statistical dynamics

0.14

0.12

0.1

Fp

0.08

0.06

0.04

0.02

0 0.5 1 1.5

v

Figure 12.8 Pinning force (in units of ν0 rp ’2 ) on a single vortex as a func-

1/2

tion of velocity (in units of · ’1 rp ’2 ν0 ) obtained from the self-consistent the-

1/2

ory for various strengths of the Hall force. The di¬erent curves correspond to

±/· = 0, 0.2, 0.4, 0.6, 0.8, 1, where the uppermost curve corresponds to ± = 0

’1/2 1/2 ’1

(m = 0.1· 2 rp 3 ν0 and T = 0.1ν0 rp ).

0.15

0.1

Fp

0.05

0

0 0.5 1 1.5

v

Figure 12.9 Pinning force (in units of 10’4 ν0 rp ’2 ) on a single vortex as a function

1/2

of velocity. Comparison of the simulation results and the results of the self-consistent

and lowest order perturbation theory, Eq. (12.165), for the case of no Hall force,

’1/2

± = 0, and a moderately strong Hall force, ± = · (m = 0.1· 2 rp 3 ν0 and T =

1/2 ’1

0.1ν0 rp ). The solid line represents the self-consistent result and the crosses the

simulation result, while the upper dash-dotted line represents the perturbation theory

result, all for the case ± = 0. The dashed line and the plus symbols represent the

self-consistent and simulation results, while the lower dash-dotted line represents the

perturbation theory result, all for the case ± = ·.

12.5. Single vortex 485

In Figure 12.9 we compare the pinning force obtained from the self-consistent

theory with the result of perturbation theory valid at high velocities, Eq. (12.165),

and simulations. According to Figure 12.9, the reduction of the pinning force due to

the Hall force predicted by the self-consistent and the perturbation theory is in good

agreement at high velocities. The pinning force obtained from the self-consistent

theory and the simulations are also in good agreement in the presence of a Hall force,

even at lower velocities. In fact in much better agreement than in the absence of

the Hall force, in accordance with the fact that the Hall force suppresses the velocity

¬‚uctuations, as we demonstrate shortly.

The Hall angle calculated from the self-consistent theory approaches from below

the disorder-independent value arctan(±/·) at high velocities, as shown in Figure

12.10.

0.8

0.7

0.6

0.5

θ

0.4

0.3

0.2

0.1

0 0.5 1 1.5

v

Figure 12.10 Hall angle as a function of velocity for a single vortex. The curves

represent the self-consistent results for the three temperatures T = 0, 0.1, 1 (in units

1/2 ’1

of ν0 rp ), where the uppermost curve corresponds to the highest temperature. The

’1 1/2

plus symbols represent the simulation results for the temperature T = 0.1ν0 rp .

The parameter ±/· is one and m = 0.1· 2 rp ν0 1/2 .

3

In Figure 12.10, the Hall angle obtained from the self-consistent theory is also

compared with simulations, and the agreement is seen to be good. As apparent from

Figure 12.10, an increase in the temperature increases the Hall angle at low velocities,

because the e¬ective friction coe¬cient decreases with increasing temperature, and

this feature vanishes at high velocities. From Figure 12.10 we can also infer the

following behavior of the Hall angle at zero velocity: at low temperatures it is zero,

since the dependence of the pinning force at low velocities is sub-linear. At a certain

temperature, the Hall angle at zero velocity jumps to a ¬nite value, since the pinning

force then depends linearly on the velocity, and saturates at high temperatures at

the disorder independent value.

486 12. Classical statistical dynamics

We have also determined the in¬‚uence of the Hall force on the velocity ¬‚uctuations

as shown in Figure 12.11.

1.04

1.035

1.03

1.025

(x ’ v)2

1.02

™

1.015

1.01

1.005

1

0 0.5 1 1.5

v

Figure 12.11 Dependence of the single vortex velocity ¬‚uctuations (in units of

· ’2 rp ν0 ) on the average velocity (in units of · ’1 rp ν0 ) for ± = · and ± = 0

’4 ’2 1/2

’1/2

1/2

(T = 0.1ν0 /rp and m = 0.1· 2 rp ν03

). The solid and dashed lines represent the

longitudinal and transverse velocity ¬‚uctuations, respectively, calculated by using the

self-consistent theory for the case ± = ·, and the plus symbols and crosses represent

the corresponding simulation results. The two dash-dotted lines represent the lon-

gitudinal and transverse velocity ¬‚uctuations, respectively, calculated by using the

self-consistent theory in the absence of the Hall force, ± = 0, which were compared

with simulations in Figure 12.7.

We observe that the Hall force at low velocities slightly increases the transverse

velocity ¬‚uctuations, and decreases the longitudinal ¬‚uctuations, whereas the longi-

tudinal and transverse velocity ¬‚uctuations are strongly suppressed by the Hall force

at higher velocities, in particular the longitudinal ¬‚uctuations. The suppression of

the velocity ¬‚uctuations is caused by the blurring by the Hall force of the preferred

direction of motion due to the external force, resulting in a less jerky motion. At

high average velocity, the longitudinal and transverse velocity ¬‚uctuations saturate

and become equal because of the strong friction. As previously discussed in the

12.6. Vortex lattice 487

absence of the Hall force, the saturation value can be determined from the energy

conservation relation (which take the same form, Eq. (12.159), as in the absence of

the Hall force, since the Hall force does not perform any work) and the high-velocity

expression for the pinning force, Eq. (12.165), since our numerical results show that

the second term on the left-hand side of Eq. (12.158) is independent of the Hall force

and velocity (and disorder). This observation tells us that the suppression of the

velocity ¬‚uctuations caused by the Hall force, according to the energy conservation

relation, Eq. (12.158), is in correspondence with the suppression of the pinning force.

We note from Figure 12.11 that the high-velocity regime sets in at lower velocities

than in the absence of the Hall force. In Figure 12.11, the velocity ¬‚uctuations cal-

culated by using the self-consistent theory are also compared with simulations, and

the agreement is seen to be good.

We have ascertained the convergence of the numerical iteration process by testing

that the obtained solutions satisfy the energy conservation relation. We ¬nd that the

energy conservation relation is ful¬lled within an accuracy of 2%, except at the lowest

velocities.

12.6 Vortex lattice

After having gained con¬dence in the Hartree approximation studying the case of a

single vortex, we consider in this section the in¬‚uence of pinning on a vortex lattice

in the ¬‚ux ¬‚ow regime, where the lattice moves with a constant average velocity,

™

uRt = v, since the external force is assumed independent of time. We consider a

triangular Abrikosov vortex lattice, and treat the interaction between the vortices in

the harmonic approximation. The free retarded Green™s function of the vortex lattice

eb (q) eb (q)

R

Dqω = (12.167)

+ i·ω ’ Kb (q)

mω 2

b

is obtained by diagonalizing the dynamic matrix, and inverting the inverse free re-

tarded Green™s function speci¬ed by Eq. (12.84) (for the moment we neglect the Hall

force). The sum in Eq. (12.167) is over the two modes, b = 1, 2, corresponding to

eigenvectors eb (q) and eigenvalues Kb (q), respectively. The eigenvalues and eigen-

vectors of the dynamic matrix are periodic with respect to translations by reciprocal

lattice vectors.

Since the lattice distortions of interest are of small wave length compared to

the lattice constant, the dynamic matrix of the vortex lattice is speci¬ed by the

continuum theory of elastic media, i.e. through the compression modulus c11 and

the shear modulus c66 , and in accordance with reference [136],

(c11 ’ c66 )qx qy

2 2

φ0 c11 qx + c66 qy

¦q = , (12.168)

(c11 ’ c66 )qx qy 2 2

c66 qx + c11 qy

B

488 12. Classical statistical dynamics

where q belongs to the ¬rst Brillouin zone, and B is the magnitude of the external

magnetic ¬eld, and φ0 /B is therefore equal to the area, a2 , of the unit cell of the

vortex lattice. In the continuum limit we obtain a longitudinal branch, el (q) · q = 1,

ˆ

with corresponding eigenvalues Kl (q) = c11 a q , and a transverse branch, et (q) · q =

22 ˆ

22

0, with corresponding eigenvalues Kt (q) = c66 a q .

12.6.1 High-velocity limit

√ 2

At high velocities, v ν0 /(·rp ), where lowest-order perturbation theory in the

disorder is valid, we can neglect the self-energies in the self-consistent expression for

the pinning force, Eq. (12.133), i.e. we can insert the free retarded Green™s function

for the lattice and, assuming v T /(·rp ), neglect the ¬‚uctuation exponent, and

obtain for the pinning force

·k · v (k · eb (k))2

dk

Fp = ’ k ν(k) . (12.169)

(·k · v)2 + (Kb (k))2

(2π)2

b=l,t

The maximum values, attained at the boundaries of the Brillouin zones, of the trans-

verse and longitudinal eigenvalues are speci¬ed by the compression and shear moduli,

Kt ∼ c66 and Kl ∼ c11 . The compression modulus is much greater than the shear

modulus, c11 c66 , in thin ¬lms and high-temperature superconductors (see for ex-

ample reference [137]). The order of magnitude of the ¬rst term in the denominator

’2

of Eq. (12.169) is ·v 2 rp , since the range of the impurity correlator is rp , and at

intermediate velocities, c66 rp /· v c11 rp /·, only the transverse mode therefore

contributes to the pinning force, and we obtain

(k · et (k))2

dk

=’ k ν(k) . (12.170)

Fp

·k · v

(2π)2

The eigenvalues et (k) are periodic in the reciprocal lattice and, assuming short-range

disorder, rp a, the rest of the integrand is slowly varying, and we obtain for the

pinning force

dk ν(k)k 2

1 ν0

Fp = ’ =’ v. (12.171)

k

(2π)2 ·k · v 8πrp ·v 2

4

2

At very high velocities, v c11 rp /·, the eigenvalues of the dynamic matrix in

Eq. (12.169) can be neglected compared with the velocity-dependent term in the

denominator, and the longitudinal and transverse parts of the free retarded Green™s

function give equal contributions to the pinning force, and we obtain

ν0

Fp = ’ v. (12.172)

4πrp ·v 2

4

12.6. Vortex lattice 489

This result is identical to the expression for the pinning force on a single vortex,

√ 2

Eq. (12.142), in the high velocity regime, v ν0 /(·rp ), since the in¬‚uence of the

elastic interaction is negligible.

12.6.2 Numerical results

In this section we consider the pinning force on the vortex lattice obtained from

the self-consistent theory. For any given average velocity of the lattice, the coupled

equations of Green™s functions and self-energies, Eq. (12.124) and Eq. (12.125), may

be solved numerically by iteration. In order to simplify the numerical calculation,

the self-consistent equations are brought on dimensionless form by introducing the

1/2 1/2

following units for length, time, and mass, a, ·a3 /ν0 , and · 2 a4 /ν0 . Starting

by neglecting the self-energies, we obtain numerically the response and correlation

functions. From Eq. (12.133) we can then determine the pinning force as a function

of the velocity. We have calculated the velocity dependence of the pinning force for

vortex lattices of sizes 4 — 4, 8 — 8, and 16 — 16 using the self-consistent theory, and

the results are shown in Figure 12.12.

5.5

5

4.5

Fp

4

3.5

3

2.5

10 20 30 40 50

v

Figure 12.12 Pinning force (in units of ν0 a’2 ) as a function of velocity (in units of

1/2

· ’1 ν0 a’2 ) obtained from the self-consistent theory for three di¬erent lattice sizes.

1/2

The stars correspond to a 4 — 4 lattice, and the two curves correspond to 8 — 8 and

16 — 16 lattices, respectively. The mass and temperature are chosen to be zero, and

1/2 1/2

the elastic constants are given by c66 a3 = 100ν0 and c11 a3 = 104 ν0 , and the

range of the disorder correlator is chosen to be rp = 0.1a.

490 12. Classical statistical dynamics

The di¬erence between the results obtained for the 8 — 8 and the 16 — 16 lattice

is small, and we conclude that the pinning force is fairly insensitive to the size of the

lattice.

In Figure 12.13 we compare the pinning force as a function of the velocity for

lattices of di¬erent sti¬nesses, and we ¬nd that the pinning force decreases with

increasing sti¬ness of the lattice.

6

5

4

Fp

3

2

1

0 10 20 30 40 50

v

Figure 12.13 Pinning force (in units of ν0 a’2 ) on a vortex lattice of size 16 — 16

1/2

as a function of velocity (in units of · ’1 ν0 a’2 ) obtained from the self-consistent

1/2

1/2

theory for the compression modulus given by c11 a3 = 104 ν0 and three di¬erent

1/2 1/2

shear moduli: c66 a3 = 50ν0 (upper dashed line), c66 a3 = 100ν0 (solid line) and

1/2

c66 a3 = 200ν0 (lower dashed line). The mass and temperature are both chosen to

be zero, and rp = 0.1a.

Generally, the interaction between the vortices lowers the pinning force, since the

neighboring vortices in a moving lattice drag a vortex over the potential barriers.

This can be inferred from the self-consistent theory by comparing the pinning forces

depicted in Figures 12.3 and 12.12, and in perturbation theory by noting the extra

term originating from the elastic interaction in the denominator of the expression for

12.6. Vortex lattice 491

the pinning force, Eq. (12.169).

When the temperature is increased, the pinning force decreases, except at very

high velocity, as apparent from Figure 12.14. This feature is common to the single

vortex case, and simply re¬‚ects that thermal noise helps a vortex over the potential

barriers.

3.7

3.6

3.5

3.4

Fp

3.3

3.2

3.1

3

2.9

10 20 30 40 50

v

Figure 12.14 Pinning force (in units of ν0 a’2 ) on a vortex lattice of size 16—16 as

1/2

a function of velocity (in units of · ’1 ν0 a’2 ) obtained from the self-consistent theory

1/2

1/2

for two di¬erent temperatures. The elastic constants are given by c66 a3 = 100ν0

’1/2

and c11 a3 = 104 ν0 , and rp = 0.1a and m = 1.0 · 10’4 · 2 a3 ν0

1/2

. The dashed line

1/2 ’1

corresponds to T = 0, and the solid line to T = 0.5ν0 a .

The convergence of the iterative procedure is monitored by checking that energy

conservation is ful¬lled. The energy conservation relation for a vortex lattice is

obtained as in Section 12.5.5, and since the term originating from the harmonic

interaction between the vortices disappears owing to the symmetry property of the

dynamic matrix, Eq. (12.76), we obtain for a vortex lattice the energy conservation

relation

· ‚t tr ’i‚t GK (R, t; R, t ) + 2 T GR (R, t; R, t ) |t =t = ’v · Fp . (12.173)

492 12. Classical statistical dynamics

The convergence of the iteration procedure, employed when solving the self-consistent

equations, has been checked by numerically calculating the terms in Eq. (12.173). We

¬nd that the right- and left-hand sides of the energy conservation relation di¬er by

no more than a few percent after twenty iterations.

12.6.3 Hall force

We now consider the in¬‚uence of a Hall force on the dynamics of a vortex lattice. The

motion of the vortex lattice, with its associated magnetic ¬eld, induces an average

electric ¬eld. The relationship between the average vortex velocity and the induced

electric ¬eld, E = v — B, and the expression for the Lorentz force, yields for the

resistivity tensor of a superconducting ¬lm

φ0 B ·e¬ ±

ρ= , (12.174)

’±

2 ·e¬

·e¬ + ±2

where the e¬ective friction coe¬cient, ·e¬ , was introduced in Eq. (12.79).13 Accord-

ing to Eq. (12.174), the following relationship between the transverse, ρxy , and the

longitudinal resistivities, ρxx , is obtained

±2

±

ρxy = ρ2 1+ . (12.175)

xx 2

Bφ0 ·e¬

If the Hall force is small, ± ·e¬ , the scaling relation

±

ρxy = ρ2 (12.176)

xx

Bφ0

is seen to be obeyed. This scaling law is valid for all velocities of the vortex, provided

the Hall force is small compared with the friction force, ± ·. We note that the

scaling law is also valid at small vortex velocities for arbitrary values of the Hall

force, if the e¬ective friction coe¬cient diverges at small velocities. This occurs if

the pinning force decreases slower than linearly in the vortex velocity. This is indeed

the case, according to the self-consistent theory, at temperatures lower than the

√

average barrier height, T ν0 /rp , as indicated by the low velocity behavior of the

pinning force in Figure 12.15. This behavior of the pinning force is also obtained for

non-interacting vortices as apparent from Figure 12.8.

In Figure 12.15 is shown the pinning force obtained from the self-consistent theory

as a function of velocity for the case of zero temperature. As expected there is

no in¬‚uence of the Hall force on the pinning force at low velocities, but we ¬nd

a suppression at intermediate velocities, and at very high velocities, v c11 a/·,

we recover the high velocity limit of the single vortex result, i.e. Eq. (12.165). By

comparison of Figures 12.8 and 12.15, we ¬nd that the Hall force has a much weaker

in¬‚uence at intermediate velocities on the pinning of an interacting vortex lattice

than on a system of non-interacting vortices. Furthermore, the in¬‚uence of the Hall

force on the pinning force is more pronounced for a sti¬ lattice than a soft lattice, as

seen from the inset in Figure 12.15.

13 The e¬ective friction coe¬cient was determined to lowest order in the disorder in reference [138].

12.7. Dynamic melting 493

3.6

3.5

3.4

3.3

3.2

Fp 1.6

1.5

3.1 Fp

1.4

3 1.3

2.9 1.2

0 50

2.8

v

2.7

0 10 20 30 40 50

v

Figure 12.15 Pinning force (in units of ν0 a’2 ) on a vortex lattice of size 16 — 16

1/2

as a function of velocity (in units of · ’1 ν0 a’2 ) obtained from the self-consistent

1/2

theory. The solid and dashed lines correspond to ± = 0 and ± = ·, respectively.

The temperature and mass are both chosen to be zero, and rp = 0.1a. The elastic

1/2 1/2

constants are given by c11 a3 = 104 ν0 and c66 a3 = 100ν0 . Inset: pinning force as

1/2

a function of velocity for ± = 0 and ± = ·, respectively. Here c66 a3 = 300ν0 and

the other parameters are unchanged.

In Figure 12.16 the dependence of the Hall angle on the velocity is presented for

various sti¬nesses of the vortex lattice; the sti¬est lattice has the greatest Hall angle.

Since the pinning force is reduced by the interaction between the vortices, the Hall

angle for a lattice is larger than for an independent vortex, except at high velocities

where they saturate at the same value. A similar behavior of the Hall angle at zero

velocity, as observed for a single vortex in Section 12.5.5, pertains to a vortex lattice.

12.7 Dynamic melting

In this section we consider the in¬‚uence of quenched disorder on the dynamic melting

of a vortex lattice. This non-equilibrium phase transition has been studied experi-

mentally [139, 140, 141, 142, 143, 144], as well as through numerical simulation and

a phenomenological theory and perturbation theory [145, 146, 147]. The notion of

dynamic melting refers to the melting of a moving vortex lattice where, in addition to

the thermal ¬‚uctuations, ¬‚uctuations in vortex positions are induced by the disorder.

A temperature-dependent critical velocity distinguishes a transition between a phase

494 12. Classical statistical dynamics

0.8

0.7

0.6

θ

0.5

0.4

0.3

0 10 20 30 40 50

v

Figure 12.16 Hall angle obtained from the self-consistent theory for a vortex lattice

of size 16 — 16 as a function of velocity (in units of · ’1 ν0 a’1 ) for a moderately

1/2

1/2

strong Hall force, ± = ·. The compression modulus is given by c11 a3 = 104 ν0 ,

and the three curves correspond to decreasing values of the shear modulus c66 a3 =

1/2 1/2 1/2

200ν0 , 100ν0 , 50ν0 . The mass and temperature are both chosen to be zero, and

rp = 0.1a.

where the vortices form a moving lattice, the solid phase, and a vortex liquid phase.

Before solving the self-consistent equations by numerical iteration in order to

obtain the phase diagram, we consider the heuristic argument for determining the

phase diagram for dynamic melting of a vortex lattice presented in reference [145].

There, the disorder induced ¬‚uctuations were estimated by considering the correlation

function

(p)

(p)

κ±± (x, t) = f± (x, t) f± (0, 0) (12.177)

of the pinning force density

f (p) (x, t) = ’ δ(x ’ R ’ uRt ) ∇V (x ’ vt) . (12.178)

R

Neglecting the interdependence of the ¬‚uctuations of the vortex positions and the

¬‚uctuations in the disorder potential, the pinning force correlation function factorizes

δ(x ’ R ’ uRt ) δ(R ’ uR 0 ) ∇±∇± V (x ’ vt)V (0) .

κ±± (x, t)

RR

(12.179)

Introducing the Fourier transform (A is the area of the ¬lm)

CRR (q, t) = A’1 e’iq·(R+uRt ’R ’uR 0 ) (12.180)

12.7. Dynamic melting 495

of the vortex density-density correlation function

δ(x ’ R ’ uRt ) δ(R ’ uR 0 )

CRR (x, t) = (12.181)

and employing the translational invariance yields

’nV δ(x ’ R ’ uRt ’ R ’ uR 0 ) ∇± ∇± ν(x ’ vt) ,

κ±± (x, t) =

RR

(12.182)

where nV is the density of vortices. In the ¬‚uidlike phase the motion of di¬erent

vortices is incoherent and the o¬-diagonal terms, R = R , can be neglected yielding

κ±± (x, t) = ’nV δ(x) ∇± ∇± ν(vt) . (12.183)

In analogy with the noise correlator, the e¬ect of disorder-induced ¬‚uctuations is

then represented by a shaking temperature

∞

1 1 ν0 1 ν0

dx dt κ±± (x, t) √ =√

Tsh = , (12.184)

3 3

4·nV 4 2π ·vrp 4 2π F rp

’∞

±

where in the last equality it is assumed that the pinning force is small compared

with the friction force, i.e. ·v F . An e¬ective temperature is then obtained by

adding the shaking temperature to the temperature, Te¬ = T + Tsh , and according to

Eq. (12.184) the e¬ective temperature decreases with increasing external force, i.e.

with increasing average velocity of the vortices. As the external force is increased the

¬‚uid thus freezes into a lattice. The value of the external force for which the moving

lattice melts, the transition force Ft , is in this shaking theory de¬ned as the value for

which the e¬ective temperature equals the melting temperature, Tm , in the absence

of disorder

Te¬ (F=Ft ) = Tm (12.185)

and has therefore in the shaking theory the temperature dependence

ν0

√

Ft (T ) = (12.186)

4 2πrp (Tm ’ T )

3

for temperatures below the melting temperature of the ideal lattice. We note that

the transition force for strong enough disorder exceeds the critical force for which

1/2 2

the lattice is pinned Ft > Fc ∼ ν0 /rp .

We now describe the calculation within the self-consistent theory of the physical

quantities of interest for dynamic melting. The conventional way of determining a

melting transition is to use the Lindemann criterion, which states that the lattice

melts when the displacement ¬‚uctuations reach a critical value u2 = c2 a2 , where cL

L

is the Lindemann parameter, which is typically in the interval ranging from 0.1 to 0.2,

and a2 is the area of the unit cell of the vortex lattice. In two dimensions the position

¬‚uctuations of a vortex diverge even for a clean system, and the Lindemann crite-

rion implies that a two-dimensional vortex lattice is always unstable against thermal

¬‚uctuations. However, a quasi-long-range translational order persists up to a certain

496 12. Classical statistical dynamics

melting temperature [146]. As a criterion for the loss of long-range translational

order a modi¬ed Lindemann criterion involving the relative vortex ¬‚uctuations

(u(R + a0 , t) ’ u(R, t))2 = 2c2 a2 , (12.187)

L

where a0 is a primitive lattice vector, has successfully been employed [146], and

its validity veri¬ed within a variational treatment [148]. The relative displacement

¬‚uctuations of the vortices are speci¬ed in terms of the correlation function according

to

(u(R+a0 , t) ’ u(R, t))2 = 2itr GK (0, 0) ’ GK (a0 , 0) , (12.188)

where the translation invariance of the Green™s functions has been exploited. The

correlation function is determined by the Dyson equation, Eq. (12.147), where the in-

¬‚uence of the quenched disorder appears explicitly through ΣK and implicitly through

ΣR and ΣA in the retarded and advanced response functions. Furthermore, the self-

energies depend self-consistently on the response and correlation functions. We have

calculated numerically the Green™s functions and self-energies and thereby the vortex

¬‚uctuations for a vortex lattice of size 8 — 8, and evaluated the pinning force from

Eq. (12.133).

We determine the phase diagram for dynamic melting of the vortex lattice by cal-

culating the relative displacement ¬‚uctuations for a set of velocities, and interpolate

to ¬nd the transition velocity, vt , i.e. the value of the velocity at which the ¬‚uctua-

tions ful¬ll the modi¬ed Lindemann criterion (the determination of the Lindemann

parameter is discussed shortly). An example of such a set of velocities is presented

in the lower inset in Figure 12.17, where the relative displacement ¬‚uctuations as a

function of velocity are shown. The magnitude of the transition force is determined

by the averaged equation of motion

Ft = ·vt + Fp (vt ) (12.189)

and is then obtained by using the numerically calculated pinning force. Repeating the

calculation of the transition force for various temperatures determines the melting

curve, i.e. the temperature dependence of the transition force, Ft (T ), separating two

phases in the F T -plane: a high-velocity phase where the vortices form a moving solid

when the external force exceeds the transition force, F > Ft (T ), and a liquid phase

for forces less than the transition force.

In order to be able to compare the results of the self-consistent theory with the

simulation results, we use the same parameters as input to the self-consistent theory

as used in the literature [145]. There, the melting temperature in the absence of

disorder is given by Tm = 0.007 (the unit of energy per unit length is taken to be

2(φ0 /4π»)2 ) as obtained by simulations of clean systems [149], and assumed equal

to the Kosterlitz“Thouless temperature [150, 151]

c66 a2

TKT = . (12.190)

4π

The shear modulus is therefore determined to have the value c66 = 0.088 (as a is taken

as the unit of length). The range of the vortex interaction, », was approximately

12.7. Dynamic melting 497

equal to the lattice spacing, a0 , giving for the compression modulus [130]

16π»2 c66

c11 = 50 c66 4.4 . (12.191)

a2

0

The range and strength of the disorder correlator in the simulations are in the chosen

units, rp = 0.2 and ν0 = 1.42 · 10’5 , and since the simulations are done for an over-

damped system, the vortex mass in the self-consistent theory should be set to zero.

As described above, our numerical results for the relative displacement ¬‚uctu-

ations can be used to obtain the dynamic phase diagram once the Lindemann pa-

rameter is determined. In order to do so we calculate melting curves by using the

self-consistent theory for a set of di¬erent values of the Lindemann parameter. We

¬nd that these curves have the same shape, close to the melting temperature, as the

melting curve obtained from the shaking theory, Eq. (12.186),

C2

T = C1 ’ . (12.192)

Ft

The curve which intersects at the melting temperature Tm = 0.007, the one depicted

in the upper inset in Figure 12.17, i.e. the one for which C1 is closest to the value

0.007, is then chosen, determining the Lindemann parameter to be given by the value

cL = 0.124.

Having determined the Lindemann parameter, we can determine the melting

curve, and the corresponding phase diagram obtained from the self-consistent theory

is shown in Figure 12.17. The simulation results of reference [145] are also presented,

as well as the melting curve obtained from the shaking theory. We note the agree-

ment of the simulation with the self-consistent theory, as well as with the shaking

theory, although the simulation data are not in the large-velocity regime and the

shaking argument is therefore not a priori valid.

In view of the good agreement between the self-consistent theory, the shaking

theory and the simulation, and the fact that we have only one ¬tting parameter at

our disposal, the melting temperature in the absence of disorder, it is of interest

to recall that while the melting curve obtained from the shaking theory was based

on an argument only valid in the liquid phase, i.e. freezing of the vortex liquid was

considered, the melting curve we obtained from the self-consistent theory is calculated

in the solid phase, i.e. we consider melting of the moving lattice. Furthermore, the

melting of the vortex lattice was indicated in the simulation by an abrupt increase

in the structural disorder [145], yet another melting criterion, and the agreement of

the self-consistent theory with the simulation data are therefore further validating

the use of the modi¬ed Lindemann criterion.

As is apparent from the upper inset in Figure 12.17, the critical exponent obtained

from the self-consistent theory, 1.0, is in excellent agreement with the prediction of

the shaking theory, where the critical exponent equals one. Furthermore, we ¬nd

that the self-consistent theory yields the value 1.65 · 10’4 for the magnitude of the

√

slope C2 , which is in good agreement with the value, ν0 /(4 2πrp ) = 1.77 · 10’4 ,

3

predicted by the shaking theory, represented by the lower dashed line. That the

values are so close testi¬es to the appropriateness of characterizing the disorder-

induced ¬‚uctuations e¬ectively by a temperature.

498 12. Classical statistical dynamics

Figure 12.17 Phase diagram for the dynamic melting transition. The melting curve

separates the two phases “ for values of the external force larger than the transition

force the moving vortices form a solid, and for smaller values they form a liquid.

The dots in the boxes represent points on the melting curve obtained from the self-

consistent theory using a vortex lattice of size 8 — 8, while the three stars represent

the simulation results of reference 6. The crosses represent the lowest-order pertur-

bation theory results. The dashed line is the curve Ft (T ) = 1.77 · 10’4 /(0.007 ’ T ),

the melting curve predicted by the shaking theory. Upper inset: relationship between

temperature and the inverse transition force obtained from the self-consistent theory,

close to the melting temperature, for the particular value of the Lindemann parame-

ter cL = 0.124, for which the curve intersects the vertical axis at Tm = 0.00701. The

set of points calculated from the self-consistent theory (plus signs) coincides with

a straight line in excellent agreement with the prediction for the critical exponent

by the shaking theory being 1. Lower inset: relative displacement ¬‚uctuations as a

function of velocity. The dots to the left are calculated by using the self-consistent

theory and the dots to the right are calculated by using lowest-order perturbation

theory (for the temperature T = 0.0065).

12.7. Dynamic melting 499

It is of interest to compare the melting curves obtained from the self-consistent

theory and perturbation theory. Expanding the kinetic component of the Dyson

equation, Eq. (12.124), to lowest order in the disorder we obtain

Dqω ΣK(1) ’ 2i·T Dqω

GK(1) R A

=

qω qω

’ 2i·kB T Dqω ΣR(1) Dqω + Dqω ΣA(1) Dqω ,

R R A A

(12.193)

qω qω

where ΣR(1) , ΣA(1) and ΣK(1) are the lowest-order approximations of the self-energies,

i.e. calculated to ¬rst order in ν0 . The relative vortex displacement ¬‚uctuations,

Eq. (12.188), can then be obtained in perturbation theory from Eq. (12.193). In Fig-

ure 12.17 is shown the melting curve predicted by perturbation theory, i.e. where for

the transition velocity interpolation we use the relative vortex ¬‚uctuations obtained

from perturbation theory, an example of which is shown in the lower inset. As is

to be expected, the perturbation theory result is in good agreement with the self-

consistent theory, and the shaking theory, at high velocities. However, we observe

from Figure 12.17 that the melting curve obtained from lowest-order perturbation

theory deviates markedly at intermediate velocities from the prediction of the non-

perturbative self-consistent theory, and thereby from the shaking theory, which is

known to account well for the measured melting curve [142].

The shaking theory is seen to be in remarkable good agreement with the self-

consistent theory for the parameter values considered above. We have investigated

whether this feature persists for stronger disorder. As apparent from Figure 12.18,

there is a more pronounced di¬erence between the shaking theory and the self-

consistent theory at stronger disorder. Whereas the deviation between the self-

consistent and shaking theory for the previous parameter values typically is 5%,

in the case of a ¬ve-fold stronger disorder, ν0 = 7.1 · 10’5 , it is more than 15%.

We have studied the in¬‚uence of pinning on vortex dynamics in the ¬‚ux ¬‚ow

regime. A self-consistent theory for the vortex correlation and response functions

was constructed, allowing a non-perturbative treatment of the disorder using the

powerful functional methods of quantum ¬eld theory presented in Chapter 10. The

validity of the self-consistent theory was established by comparison with numerical

simulations of the Langevin equation.

The self-consistent theory was ¬rst applied to a single vortex, appropriate for

low magnetic ¬elds where the vortices are so widely separated that the interaction

between them can be neglected. The result for the pinning force was compared with

lowest-order perturbation theory and good agreement was found at high velocities,

whereas perturbation theory failed to capture the non-monotonic behavior at low

velocities, a feature captured by the self-consistent theory. The in¬‚uence of the Hall

force on the pinning force on a single vortex was then considered using the self-

consistent theory. The Hall force was observed to suppress the pinning force, an

e¬ect also con¬rmed by our simulations. The suppression of the pinning force was

shown at high velocities to be in agreement with the analytical result obtained from

lowest-order perturbation theory. The suppression of the pinning force was caused

by the Hall force through its reduction of the response function, while the e¬ect of

¬‚uctuations through the ¬‚uctuation exponent at not too high temperatures could be

500 12. Classical statistical dynamics

6

5

4

Ft 3

2

1

0

0.006 0.0065 0.007

T

Figure 12.18 Phase diagram for the dynamic melting transition for the disorder

strength ν0 = 7.1 · 10’5 . The plus signs represent points on the melting curve

obtained from the self-consistent theory for a vortex lattice of size 8 — 8, while the

dashed curve is the curve Ft (T ) = 8.85 ·10’4 /(0.007’T ), the melting curve predicted

by the shaking theory.

neglected. The situation at high temperatures was the opposite, since in that case

the thermal ¬‚uctuations were of importance, and the Hall force then increased the

pinning force because it suppressed the ¬‚uctuation exponent.

We also studied a vortex lattice treating the interaction between the vortices in

the harmonic approximation. The pinning force on the vortex lattice was found to

be reduced by the interaction. The pinning force as a function of velocity displayed a

plateau at intermediate velocities, before eventually approaching at very high veloci-

ties the pinning force on a single vortex. Analytical results for the pinning force were

obtained in di¬erent velocity regimes depending on the magnitude of the compression

modulus of the vortex lattice. Furthermore, we included the Hall force and showed

that its in¬‚uence on the pinning force was much weaker on a vortex lattice than on

a single vortex.

We developed a self-consistent theory of the dynamic melting transition of a vortex

lattice, enabling us to determine numerically the melting curve directly from the

dynamics of the vortices. The presented self-consistent theory corroborated the phase

diagram obtained from the phenomenological shaking theory far better than lowest-

order perturbation theory. The melting curve obtained from the self-consistent theory

was found to be in good quantitative agreement with simulations and experimental

data.

12.8 Summary

In this chapter we have considered the theory of classical statistical dynamics treating

systems coupled to a heat bath and classical stochastic forces. In particular we

12.8. Summary 501

studied Langevin dynamics and quenched disorder, and applied the method to study

the dynamics of the Abrikosov ¬‚ux line lattice. As to be expected, the formalism of

classical statistical dynamics is the classical limit of the general formalism of non-

equilibrium states, Schwinger™s closed time path formulation of quantum statistical

mechanics, the general technique to treat non-equilibrium states we have developed

and applied in this book. The language of quantum ¬eld theory is thus the tool to

study ¬‚uctuations whatever their nature might be.

Appendix A

Path integrals

Quantum dynamics was stated in Chapter 1 in terms of operator calculus, viz.

through the Schr¨dinger equation or equivalently via the Hamiltonian as in the evo-

o

lution operator. Alternatively, quantum dynamics can be expressed in terms of path

integrals which directly exposes the basic principle of quantum mechanics, the su-

perposition principle1 . To acquaint ourselves with path integrals we show here for

the case of a single particle the equivalence of the two formulations by deriving the

path integral formulation from the operator expression for Dirac™s transformation

ˆ

function of Eq. (1.16), x, t|x , t = x|U (t, t )|x = G(x, t; x , t ) ≡ K(x, t; x , t ).

Propagating in small steps by inserting complete sets at intermediate times we have

for the propagator

dx1 dx2 . . . dxN x, t|xN , tN xN , tN |xN ’1 , tN ’1

x, t|x , t =

— xN ’1 , tN ’1 |xN ’2 , tN ’2 · · · x1 , t1 |x , t . (A.1)

We are consequently interested in the transformation function for in¬nitesimal times,

and from Eq. (1.16) we obtain

xn |e’

ˆ

i

xn , tn |xn’1 , tn’1 |xn’1

”tH(tn )

=

”t ˆ

δ(xn ’ xn’1 ) + xn |H(tn )|xn’1 + O(”t2 ) ,

= (A.2)

i

where ”t = tn ’ tn’1 = (t ’ t )/(N + 1)), as we have inserted N intermediate

resolutions of the identity.

In the following we shall consider a particle of mass m in a potential V for which

we have the Hamiltonian

p2

ˆ

ˆ

H(t) = + V (ˆ , t) , (A.3)

x

2m

ˆ pˆ

i.e. H = H(ˆ , x, t), where H by correspondence is Hamilton™s function.

1 For a detailed exposition of how the superposition principle for alternative paths leads to the

Schr¨dinger equation, we refer the reader to chapter 1 of reference [1].

o

503

504 Appendix A. Path integrals

Inserting a complete set of momentum states, we get

xn |H(ˆ , p, tn )|xn’1 xn |H(xn , p, tn )|xn’1

xˆ ˆ

=

dpn

e pn ·(xn ’xn ’1 ) H(xn , pn , tn ) ,

i

= (A.4)

d

(2π )

where we encounter Hamilton™s function on phase space

p2

n

H(xn , pn , tn ) = + V (xn , tn ) . (A.5)

2m

Inserting into Eq. (A.2), we get

dpn ”t

e pn ·(xn ’xn ’1 ) 1 +

i

xn , tn |xn’1 , tn’1 H(xn , pn , tn ) + O(”t2 )

= d

(2π ) i

dpn

e [pn ·(xn ’xn ’1 )’”tH(xn ,pn ,tn )] + O(”t2 ) .

i

= (A.6)

d

(2π )

Inserting additional internal times, we approach the limit ”t ’ 0, or equivalently

N ’ ∞, obtaining for the transformation function

N N +1

dpn

e [pn ·(xn ’xn ’1 )’”tH(xn ,pn ,tn )]

i

x, t|x , t = lim dxn

(2π )d

N ’∞

n=1 n=1

Dxt Dpt i

¯ ¯ t

dt [pt ·xt ’H(xt ,pt ,t)]

¯ ¯ ™¯ ¯ ¯¯

≡ e , (A.7)

t

d

(2π )

where x0 ≡ x , and xN +1 ≡ x. In the last equation we have just written the limit of

the sum as a path integral, and the integration measure has been identi¬ed by the

explicit limiting procedure.

The Hamilton function is quadratic in the momentum variable, and we have

Gaussian integrals which can be performed

∞ xn ’xn ’1 2

dpn p2 m d/2

xn ’xn ’1 i

e ”t(pn · ”t ’ 2m ) =

m ”t

i n

e2 (A.8)

”t

d

’∞ (2π ) 2πi ”t

and we thus get for the propagator

N +1 m (xn ’xn ’1 )2

N

’V (xn ,tn )

i

1 dxn ”t 2”t

K(x, t; x , t ) = lim e n =1

’d/2 ’d/2

N ’∞ m m

n=1 2πi ”t

2πi ”t

xt =x

t

dt L(xt ,xt ,t)

i ¯ ¯ ™¯ ¯

≡ Dxt e , (A.9)

t

¯

xt =x

Appendix A. Path integrals 505

where L in the continuum limit is seen to be Lagrange™s function

1

mx2 ’ V (xt , t) = xt · pt ’ H(xt , pt , t)

™ ™t ™

L(xt , xt , t) = (A.10)

2

related to Hamilton™s function through a Legendre transformation. The integration

measure has here been obtained for the case where we take the piecewise linear

approximation for a path.2

Instead of formulating quantum dynamics in terms of operator calculus we have

thus exhibited it in a way revealing the underlying superposition principle, viz. ac-

cording to Feynman™s principle: each possible alternative path contributes a pure

phase factor to the propagator, exp{iS/ }, where

t

¯ ¯ ™¯ ¯

S[xt ] = dt L(xt , xt , t) (A.11)

t

is the classical action expression for the path, xt , in question.3

The classical path is determined by stationarity of the action

δS

=0 (A.12)

δxt xt =xc l

t

the principle of least action,4 or explicitly through the Euler-Lagrange equations

d ‚L ‚L

’ =0 (A.13)

™

dt ‚x ‚x

the classical equation of motion.

Formulating quantum mechanics of a single particle as the zero-dimensional limit

of quantum ¬eld theory amounts to focussing on the correlation functions of, for

example, the position operator in the Heisenberg picture, say the time-ordered cor-

relation function

GH (t, t ) ≡ T (ˆ H (t) xH (t ))

ˆ (A.14)

x

where the bracket refers to averaging with respect to some state of the particle, pure