Quantum corrections to SG equation solutions and applications

Physics Letters A 376 (2012) 991–995

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Physics Letters A www.elsevier.com/locate/pla

Quantum corrections to SG equation solutions and applications Grzegorz Kwiatkowski, Sergey Leble ∗ Gda´ nsk University of Technology, ul. G. Narutowicza 11/12, 80-952 Gda´ nsk, Poland

a r t i c l e

i n f o

Article history: Received 11 September 2011 Received in revised form 5 December 2011 Accepted 18 January 2012 Available online 24 January 2012 Communicated by A.R. Bishop

a b s t r a c t Quantum corrections to classical solutions of one-dimensional Sine-Gordon model are evaluated with account of rest d − 1 dimensions of a d-dimensional theory. A quantization of the models is considered in terms of space–time functional integral. The generalized zeta-function is used to renormalize and evaluate the functional integral and quantum corrections to energy in quasiclassical approximation. The results are applied to appropriate crystal dislocation models and magnetic domain walls dynamics, which kink energy and the corrections are numerically calculated. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The history of functional integral approach in quantum physics was started from the celebrated paper of R. Feynman [1], where the Heisenberg quantization principle was expressed in terms of evaluation of admissible classical trajectories contributions. Beginning from Maslov paper [2] the functional integral method becomes a practical tool for evaluation of quasiclassical corrections to the action. At the meantime [3] such expressions were studied for Sine-Gordon (SG): approximate quantum corrections were evaluated, for early review of non-perturbative methods see [4]. The problem of embedding such model into real multidimensional theory and the problem of renormalization is still under examination [5–8]. In [6], where the general method is elaborated for arbitrary background profiles, expressions for ground state energies were evaluated for a 3 + 1 dimensions case with potential dependent on a single variable. Generalization for the supersymmetric kink is given in [9]. The aim of this work is to apply the semi-classical quantization method to calculate energy correction for Sine-Gordon kinks in real crystals. Having in mind more wide applications, including other models and periodic solutions we briefly demonstrate details of the Feynman integral construction and generalized zeta function evaluation as well as the renormalization realization. A general algebraic method of quantum corrections evaluation based on zetafunction [10,5] is used and the Green function for heat equation with elliptic potential is constructed (see also [8]). We consider Frenkel–Kontorova models for crystal structure dislocations [11,12] and its Sine-Gordon (SG) equation counterpart. Being a one-dimensional field theory, which continuous limit is

based on a version of nonlinear Klein–Fock–Gordon (KFG) equations, it should be embedded into two- or three-dimensional picture with effective account of rest variables as in the model itself as in quantization procedure. We also apply the results to the similar model of quasi-one-dimensional magnet dynamics [13]. In the first section we show some important details of a space– time approach to the continual integral evaluation originated from [1] and its semi-classical realization for static field equation’s solutions as well as the zeta-function renormalization method, taking into account both vacuum choice and ultraviolet divergence compensation. In Section 2.3 we introduce physically necessary dimensions in basic operator to evaluate its contributions to the theory (Section 2.4). In the next section a connection between Frenkel–Kontorova model for a real solid and Sine-Gordon equation is given and the kink solution of Sine-Gordon equation is reproduced within the realistic parametrization scheme. The subsections are devoted to calculating corrections to energy of Sine-Gordon kink with expressions via constants of some solids. These corrections are linked to Frenkel–Kontorova model of dislocations. The last one presents the corrections to SG kink model of the quasione-dimensional magnetic domain wall dynamics. We expect that quantum contributions would be measured in precise experiments and could be important in a framework of practical devices with precise deformation control. 2. Method examination 2.1. Functional integral Let us start with a one-dimensional nonlinear SG equation in dimensionless variables t =

*

Corresponding author. E-mail addresses: [email protected] (G. Kwiatkowski), [email protected] (S. Leble). 0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2012.01.030

ϕ = −

m2 2π

sin(2πϕ ),

t T

, x=

=

x , a

ϕ=

1 ∂2 c2 ∂ t 2

−

ϕ T ϕ0 and c = c a

∂2 ∂ x2

(1)

992

G. Kwiatkowski, S. Leble / Physics Letters A 376 (2012) 991–995

with dimensionless parameters c as propagation speed and m2 as potential amplitude. Primes mark dimensional variables, the sense of constants T , a, ϕ0 will be explained in the context of concrete models. It is the Euler equation of the variational principle with the action defined as

S (ϕ ) = T

ϕ02 G

1 dt

2



∞

dx

c2

−∞

0



1

∂ ϕ (x, t ) ∂t

2

 −

∂ ϕ (x, t ) ∂x

2

(2)

with G introduced as harmonic interaction coefficient (as in (34) for example). The sense of the scale parameter T hence becomes obvious from this definition (2). Feynman formulation of quantum field theory links classic systems with their quantum counterparts through usage of the classical action in the path integral defining the propagator [1]



i

ψ|e − h¯ T H |θ =

i

D φ(x, t ) e h¯ S (φ)

(3)

0,1

C θ,ψ 0,1

with H as quantum Hamiltonian and C θ,ψ as family of all continuous functions fulfilling boundary conditions: φ(x, 0) = θ(x), φ(x, 1) = ψ(x). The Feynman integral can be approximated by the use of stationary phase method [2] (since any classical solution (ϕ ) is a stationary point of the action functional) with limits on the spatial interval to be specified depending on a particular problem i

i

ϕ |e − h¯ T H |ϕ   e h¯ S (ϕ )

∞

−∞

−π

dak e



j ,l a j al (φl , L φ j )

,

(4)



 L = − A  + m cos 2πϕ (x) , i ϕ02 G T

A = −

2π h¯

(5)

.

It is important to note, that static solutions of the field equation ϕ are assumed to be eigenstates of the Hamiltonian corresponding to quantum energy E q . Since L is a sum of one-dimensional second-order linear Hermitian operators (t ∈ [0, 1], x ∈ −∞, ∞, zero boundary conditions), its eigenfunctions form a complete set of orthonormal functions which we choose as the set {φi } ∼ λi . After simple transformations and rescaling one arrives at

e

− h¯i

T Eq

ϕ |ϕ   e

i h¯

S (ϕ )



e

− 12

=√

λk

k

A=

A r2

, 

E q  E (ϕ ) +

2iT

ln det[ L ] −

i h¯



ϕ02 G 2

+

∞

 dx

−∞

m2 

h¯ iT

2π 2

∂ ϕ (x, t ) ∂x 

ln det[ L 0 ] −

h¯ iT

lnϕ |ϕ .

(9)



(10)

L 0 = − A .

(11)

This provides a more transparent form, when additional spatial dimensions are introduced. Second step is to rewrite the determinants in a form that realizes the subtraction (9). For this purpose, let us define generalized zeta-function (GZF) as a formal sum

ζ L (s) =



λk−s ,

(12)

k

where λk are eigenvalues of a given operator L on finite interval which dimension tends to infinity, then

ln det[ L ] = −

dζ L ds

(0).

(13)

S (ϕ )

det[ L ]

 ∂ + L g L ( y , x, x0 ) = δ( y )δ(x − x0 ). ∂y

,

lnϕ |ϕ ,

(14)

The Green function for this equation can be written (for y > 0) as

g L ( y , x, x0 ) =



e −λk y ψk (x)ψk∗ (x0 ),

(15)

k

where ψk are orthonormal eigenfunctions of the operator L. Only eigenvalues of the operator L are important for further calculations, so one can integrate over all spatial variables (the resulting function is defined as γ ):



γL ( y) =

g L ( y , x, x) dx =

(6)



e −λk y .

(16)

k

By performing Mellin transform

(7)

ζ L (s) =

1

+∞

(s)

y s−1 γ L ( y ) dy

(17)

0

where E (ϕ ) is the classical energy of static solution ϕ , for which classical action can be expressed as S (ϕ ) = − T E (ϕ ), or to be more precise

E (ϕ ) =

2iT

L = − A  + m2 cos 2πϕ (x) ,

which gives the expression for E q

h¯

h¯

ln det[ L ] −

There as yet remains a T -dependent term (divergent for both T → 0 and T → ∞), that has to be accounted for [3]. In distinction to [5] we propose to cut this divergent contribution by introducing the multiplicative renormalization term (called the mass scale in [5] and appearing in [6] as μ) directly in the operator L and L 0 instead of zeta-function:





2

h¯ 2iT

To obtain ζ functions, when spectra of involved operators are unknown, one should consider an equation [14]

k

where 

Solving the problem of infinite determinant value for unbounded spectrum of the differential operators we subtract analogous term for the vacuum operator (L 0 = − A  ) [5,8]

E q  E (ϕ ) +



  m  − 1 − cos 2πϕ (x, t ) 2π 2 2

2.2. On renormalization

2

one obtains the generalized Riemann zeta-function. For an operator that can be written as a sum of operators dependent on different variables L = L 1 + L 2 , function γ is a product of analogous functions of the summed operators [8]

γL = γL1 γL2 ,  

1 − cos 2πϕ (x, t )

.

(8)

(18)

what greatly simplifies calculations and shows a simple way of introducing additional spatial dimensions. In this work we use the following notations:

G. Kwiatkowski, S. Leble / Physics Letters A 376 (2012) 991–995

 L1 = − A − L2 = −

   ∂2 2 + m cos 2 πϕ ( x ) , ∂ x2

A ∂2 c2 ∂ t

. 2

(19)

E d =1 = −

(20)

E d =2 = −

In summary, the correction to classical solution energy is evaluated by the formula (9) via the Green function diagonal (16) integration, Mellin transformation (17) and renormalization parameter r choice. 2.3. Additional variables Many physical objects or phenomena described by a onedimensional equation are multidimensional in nature (domain walls, dislocation lines, etc.). This has usually no effect on classical solutions as long as they are uniform in those additional variables. However, it does change the quantum system significantly. Thanks to the property (18) we can easily account for any number of independent variables. For that, we introduce an additional operator

L3 =

A

(21)

d−1 ,

l2

h¯ mc  2T

π

h¯ m2 cl 4π T

993



2 − 2 ln(m) − ln(− A ) ,

(28) (29)

, 

 h¯ m3 cl2 1 2

E d =3 = 1 + − 2 ln ( m ) − ln (− A ) . 2 2T 4π

3

3

(30)

It is important to note, that the dependence on T is only apparent, since c is linearly dependent on T due to our choice of dimensionless variables. The choice of the scaling parameter r (included in A) allows to eliminate a dependence of the result on T (the simplest choice is r 2 = π h¯ε T ). We can insert propagation speed in natural units as well. The result is

E d =1 = −

E d =2 = −

E d =3 =

h¯ mc 

πa

(31)

,

h¯ m2 c l 4π a

h¯ 5m3 c l2 72π 2 a

(32)

,

(33)

.

assuming here for simplicity, that each additional dimension spans from 0 to al (with a as defined earlier and l as a dimensionless z parameter) and dimensionless variables z = la , which will in turn span on a range of [0, 1]. We also assume the same harmonic interaction coefficient G for all directions. Laplace operator covers all spatial variables except for x and d denotes total number of spatial variables.

This results with corrections coincident for 1 + 1 dimensions with one of [3], obtained with a different method. A development of the method for 3 + 1 dimensions is presented in [6], where general expressions for effective potential were given through mode summation. For general quantization of arccosh2 potentials via generalized zeta-function see [5].

2.4. Energy corrections via generalized zeta function

3. Corrections for a Sine-Gordon kink in different physical context

Following calculations from [15] we can readily write

γ1 ( y ) =

√ 1

Ay exp m2 Ay τ 2 dτ , √

2im

3.1. Dislocations

(22)

π

0

as the renormalized function

γL2 ( y) = √ 2

c

(23)

Aπ y

for L 2 . By analogy, function



γL3 =

γ for operator L 1 and



l2

γ for L 3 will take form:

d −1 2

(24)

.

−4 A π y

It is worth mentioning, that γ functions for L 2 and L 3 operators are calculated here using continuous approximation of their spectra, therefore particular choice of boundary conditions doesn’t change the results. We proceed with:

γ ( y ) = γ1 ( y )γL 2 ( y )γL 3 ( y ), γ ( y) =

imcld−1 2d−1 π

d +1 2

(25)

1

(− Ay )

d −1 2



exp −m2 (− A ) y τ 2 dτ .

(26)

0

After applying Mellin transform (convergent for Re(s) < d2 ) we obtain:

ζ (s) =

1 d −1 im−2s+d (− A )−s c (s − d− )l 2

2d−1 π

d +1 2

(s)(d − 2s)

Energy corrections will take form:

.

(27)

An outline for modeling edge dislocations through Frenkel– Kontorova model consists of two steps [12]. Firstly one simulates cross-section of the dislocation (in direction of Burgers vector) as a Frenkel–Kontorova kink. Parameters for the equation

M

d 2 xi dt 2

= G (xi +1 − 2xi + xi −1 ) −

επ a

 sin

2π x i



a

(34)

are introduced as follows: M is the mass of atoms of which the particular crystal lattice is built, a is the lattice constant, G is proportional to the bulk modulus K of a given material (G = K a) and 3

ε is proportional to shear modulus M s (ε = a2πM2s ).

At this point one can calculate energy of the dislocation and determine the Peierls–Nabarro potential [16], which is accountable for interactions between moving kinks and crystal lattice (important for dislocation’s dynamics modeling). Approximate solutions of Frenkel–Kontorova equation are often constructed on basis of Sine-Gordon soliton with the important distinction, that there are two static Frenkel–Kontorova kinks with different energies (energy difference gives the amplitude of Peierls–Nabarro potential). In the second step one describes the dislocation line as a row of independently moving Frenkel–Kontorova kinks that interact harmonically with their nearest neighbors and are subject to Peierls– Nabarro potential (or its first term in Fourier expansion). This means, that the dislocation line is described by another Frenkel– Kontorova equation, which can be well substituted by Sine-Gordon 2 equation (1) with m2 = 2aπ2 Gε2 , c 2 = 2

G2 T 2 M2

and natural normalization

ϕ0 = a (G 2 , M 2 and ε2 are calculated further in the text and analogous to G, M and ε ). It is important to note, that the harmonic interaction coefficient as well as mass is not the same as in (34).

994

G. Kwiatkowski, S. Leble / Physics Letters A 376 (2012) 991–995

Movement of a dislocation is simulated through creation and propagation of kink–antikink pairs on the dislocation line as a response to applied stress [17]. Often the above described framework is enriched by taking into account additional terms of the potential or anharmonic interactions. It might also be important to consider thermal oscillations of the lattice in modeling macroscopic properties of crystals. Peierls–Nabarro potential’s amplitude (ε2 ) and harmonic interaction coefficient (G 2 ) for dislocation line modeling were calculated numerically. Firstly approximate static Frenkel–Kontorova kinks were obtained by substituting Sine-Gordon kink as an initial condition to a critically damped Frenkel–Kontorova equation. By changing initial position of the kink, one gets two different stable Frenkel–Kontorova kinks, which correspond to minimal and maximal static kink energy. Difference of their energy gives ε2 . G 2 is estimated on assumption, that energy of any two atoms interacting is given by

C1

E (r ) =

−

r2

C2 r

(35)

,

where C 1 and C 2 are fitted so that E (a) is the minimum and E  (a) = G

⎧ ⎨ C = 1 a4 G , 1 2 ⎩ C 2 = a3 G .

(36)

Interaction energy for a given displacement between kinks ( X ) is then calculated by adding up interaction energy of atom pairs from two neighboring kinks. To avoid major rounding up inaccuracy and set in the zero for energy, one subtracts an analogous term for X = 0 on every iteration. By varying X , one can obtain G 2 by fitting G22 X 2 as the energy. It is important, to fit this function separately for both Frenkel–Kontorova kinks, which means that X = na. Effective mass of the kink is obtained thanks to the fact, thatSine-Gordon equation is Lorentz invariant (with sound a2 G M

velocity

M2 =

E0 M a2 G

used instead of speed of light in vacuum)

6

= 1

π

 M

√

2ε a2 G

(37)

E

=−

a2 G

h¯ π

2a2

√

G 2 M2

.

(38)

For an iron monocrystal this will give EE ≈ 0.013, which is, perhaps, as yet impossible do detect in direct measurements. In [15] one can find results for different metals. Conversely, model of the dislocation line itself should take it’s length into account. Thus we should consider it as a two-dimensional system and first-order energy corrections for creation of a dislocation would take form (29)

E E

=

√ ε . √

h¯ π 2 4a3 G

2M





 

S n · S n+1 + D S nz 2 − g μ B B · S n ,

n

j  S ni , S m

k = i jk S m δnm ,

(40)

with J , D and g as experimentally estimated parameters, S ni as coefficients of the spin vector of nth atom, B as magnetic flux density and μ B as Bohr magneton. Assuming x as the direction of the being perpendicular to the chain, rewriting S in spherical chain, B coordinates and taking continuum limit of the system one obtains

 



1

∂θ 2 ∂x

2

1



∂φ 2 ∂x  D 2 + cos (θ) − g μ B B S cos(φ) sin(θ) .

H = J S 2a

dz

+

2

sin2 (θ)

J

(41)

If D S  g μ B B, then the θ will be close to π2 and the equations of motion can be simplified to a Sine-Gordon equation [18]:

∂ 2φ ∂ 2φ gμB B 0 sin(φ) = 0, − 2 + 2 2a2 S 2 D J ∂ t 2 ∂x a JS π 1 ∂φ θ− = , 2 2D S ∂ t 1

(42)

with S-spin amplitude, φ -spin angle in the easy plane in radians, θ -the second spherical coordinate with 0 set in the direction of magnetic field, B 0 -external field amplitude, a-lattice constant, t in gμ B units of ( J S )−1 . This gives m2 = a2 BJ S 0 and c 2 = a2 2D J S 2 . Classical energy [13] and one-dimensional quantum corrections in this setup will take the form:

Ec = 8



E = −

J S 3 gμB B 0, h¯ 

π

2D S g μ B B 0 .

(43) (44)

It is sometimes important to consider ratio of those values

,

where E 0 = π 8ε is the energy of a static kink. Described model of dislocation movement is essentially a one-dimensional system, thus the quantum corrections should take form (28)

E

H =−J

(39)

The classical energy in such a case should be treated as per single chain of atoms, so the l coefficient vanishes. For an iron monocrystal this will give EE ≈ 0, 001, which is very small. 3.2. Domain walls For the purpose of this section we adopt notation and units of [18]. Spin structure of an atomic chain in a crystal with an easyplane anisotropy experiencing a constant external magnetic field is often modeled with a classical Hamiltonian [13]

    E  2D   = h¯ .  E  8S π J

(45)

c

If we were to account for the two additional dimensions, corrections would take form (30)

    E  5h¯ g μ B B 0 2D   .  E  = 72π 2 a2 S 2 J J c

(46)

Note, that (43) should be treated as energy per single chain of atoms, thus l2 term will vanish. It is most important, that in this case, relative quantum corrections are directly proportional to the external magnetic field. Such a phenomenon should be noticeable in a strong magnetic field. 4. Conclusion In our Letter we consider Sine-Gordon equation, but all ingredients of the theory could be applied to other models [8]. Inclusion of all due rest variables in a given model has a qualitative impact on energy of physical objects, even if classical models do not show such a behavior. In some cases a problem of measuring energy corrections could be posed. In particular, specific dependence of domain wall energy on external magnetic flux density could be important for magnetic properties of various materials.

G. Kwiatkowski, S. Leble / Physics Letters A 376 (2012) 991–995

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