Ginzburg - Landau free energy for constant magnetic field…

Our starting point is the two dimensional GL free energy:

'  ( )) _^*+ ,-. ^/_*⁰12 ,^3 4561 3 7, 8||^3^9:_^;||^< (2.1.1) here  is the order parameter of the superconductivity, the gauge =  , 0 describes a nonfluctuating constant magnetic field. > and ?⁰ are the mass and charge of the

Copper pair. 45  4_1  @ and A^B are phenomenological parameters; @ C /. Throughout most of the paper will use the following units. Unit of length is magnetic length E _√9^F , the coherence length H  I_J;^*+ _K and unit of magnetic field is _, so

that dimensionless magnetic field is A C_M^L

K+. I introduce weak so called N_ disorder by adding a space dependent contribution 7,  to the coefficient of quadratic term.

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Lowest Landau level approximation and quasimomentum basis

Expanding the field in quasimomentum basis only within the LLL¹²

,   ∑ P_{Q Q}R_S, (2.1.2) where the coefficients P_Q are complex numbers and R_S are quasi-momentum basis. We can find the property of R_S easily

R_Q  ?Yi[^_{`</sub>jR<sub></sub>6  ^<sub>_</sub>,  3 ^<sub>`}8. 2.1.4

It is convention to prove that quasi-momentum basis satisfy magnetic translations¹⁸ which is defined as

_lR_Q  ?^/SlR_Q , 2.1.5

here l is a general displacement vector and _l is magnetic translation operator.

_l  ?^h/l·no, 2.1.5

with a generator defined by

p_/  [q_/  r_s/t_s  Ŷ_/  r_s/t_s , r  v0 A0 0w , =^/  r_/st_s . r is the Landau gauge matrix.

The sample and periodic boundary conditions

we work in reciprocal lattice vector, so that S with the basis vector is

S  ^_l‚_~3 ^_l‚_.

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The quasi-momentum basis satisfy magnetic translations

_|,   ?^/Sz,  2.1.10

and then we have

?^/Sz  1

^_` 2V

} ‡<sup>`</sup>, ‡<sub>`</sub>  0, ˆ1, ˆ2, …

^_{_}  ^x_z ‡_{_}, ‡_{_}  0, ˆ1, ˆ2, … 2.1.11

Because of I used the LLL approximation, the ranges of ‡_` and ‡_{_} are

‡_`  0, 1, … , }  1

‡_{_}  0, 1, … , }  1

2.1.12

Thus, the basis satisfy magnetic translations and we have the periodic boundary condition.

2.2. Free energy expressed via quasimomentum variables. Clean case.

The GL free energy equation for pure vortex system is

'  ( )) _^*+ ,-. ^/_*⁰=2 ,^3 4_1  @||^3^9B_||^<. 2.2.1

In order to get the dimensionless LLL free energy, I rescaled ^ ‰ I_Š+9^;_:<x^、 ‰^`_Š、

 ‰ ^__Š , then we obtain

^‹

; _<x^ ( )) vW_;||^3^_||^<w. 2.2.2

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here W_{; 9}^J_:/+^Œx_;^/+_/+^Š is the reduced temperature, and 4_Ž  4_1  A  @.

I used some basic formulas¹⁸ to calculate the dimensionless GL free energy as follows.

The two function product is

( )RtR_{” S}⁰t?Y[ · ‘’  4V^∆_‘,S'‘. 2.2.3

here

'‘  ?Y v^/x_ –_^ –_w ?Y v_<9^‘+^/—_9^˜—™3^/Q˜₉^—™w. 2.2.4

and the Kronecker delta is defined by:

∆_‘,S ∆‘  S  š 1, [› ‘  S 3 –^)_3 –_)_

0, œ@?tž[Ÿ?  . 2.2.5

The momentum ‘ is composed of an “integer part”

¡  –_)_3 –_)_. 2.2.6

–_  … , 1, 0, 1, …, and –_  … , 1, 0, 1, …, which belonging to the reciprocal lattice and a “fractional part”

S  ^_)_3 ^_)_. 2.2.7

^_  0,^_z,^_z, … ,^zh_z and ^_  0,^_z,^_z, … ,^zh_z which belongs to the Brillouin zone.

The inverse Fourier transform of Eq.(2.2.3) is

RtR_S⁰t  X ?Y[S 3 ¡ · ’?Y ¢V[

2 –^3 –_£ ?Y \S 3 ¡^

¤¥#¦¤+¥#+ 4

[^_{`</sub>3 –<sub>`}6^_{_}3 –_{_}8

2 3 [^_`6^_{_}3 –_{_}8£.

2.2.8

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We turn back to calculate the terms of GL free energy.

Quadratic term

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8V È |, |1 _{`,_} ^<

1

4 }^X ÊX ?Y Z[V \ ^B2E_ E_

©

±,n

31

2 Y^3 2E_ ^BB2Y_3 _  Y_3 _’ad ?Y ¢[V 2  ^B

 ^B£ ?Y \± 3 n4 ^a P_²⁰_{¦Š’,²+¦Š+’}P_©Ë



.

2.2.15

The detail calculation of dimensionless Ginzburg Landau free energy for pure vortex system is shown by previous work. Next, we calculate the disorder term and define the dimensionless parameter ζ# which controls the relative disorder strength.

2.3 Disorder term in the GL free energy

The disorder term of GL free energy function is

( 4_{`,_} 1  @7, ||^. 2.3.1

with white noise correlator.

7, 70,0ÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌ  ÍNN. 2.3.2

Rescaling field as ^ ‰ I_Š+9^;_:<x^ and setting dimensionless lengths via  ‰^`_Š、 ‰^__Š, we obtain the disorder term of dimensionless GL free energy equation:

( Î,  ||_{`,_} ^. . 2.3.3

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Here

Î,   _x9^ŠJ;KhÏ_:;/+ 7, . . 2.3.4

According to Eq. 2.3.3, we have the following variance

Î, Î0,0ÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌ Í^BNN, 2.3.5

where Í^B v_x9^ŠJ;KhÏ_:;/+w^Í.

Representation of the random potential in terms of a complex random numbers

I used the random potential in the disorder term of GL equation as 7,   W_,,,

3 X ªW_²,,Ս,?Y[± 3 n · ’

²Ö,Ս×,²+g,Õ+g

3 W_²⁰_,,Ս,?Y[± 3 n · ’j

3 X ªW_±,n?Y[± 3 n · ’ 3 W_±,n⁰ ?Y[± 3 n · ’«

²+¦Õ+Ö

.

2.3.6

here W is the complex random numbers, W_±¦n  W_h±¦n⁰

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and its distribution is divided into five parts, see Fig. 2-3-1. The white noise correlator is 7, 70,0

ÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌ WÌÌÌÌÌÌÌÌ_,,,^

3 X ªW_²⁰_,,Ս,W_²,,Ս,?Y[± 3 n · ’

²Ö,Ս×,²+g,Õ+g

3 W_²⁰_,,Ս,W_²,,Ս,?Y[± 3 n · ’j

3 X ªW_±,n⁰ W_±,n?Y[± 3 n · ’ 3 W_±,n⁰ W_±,n?Y[± 3 n · ’«.

²+¦Õ+Ö

2.3.7

with some basic relations as follows:

W_,,,^ ÌÌÌÌÌÌÌÌ Í

2V}^,

Í?WÌÌÌÌÌÌÌ^  Ø>WÌÌÌÌÌÌÌ^{ }_xz^Ù₊  Ú^, 2.3.8

Fig. 2-3-1.

The distribution of the random potential in momentum space

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here Ú^ is the variance of the normal distribution. Substituting Eq.( 2.2.11), Eq.( 2.3.4) and Eq.(2.3.6) into Eq.(2.3.2), the disorder term of dimensionless GL free energy equation is

È Î, |, |^

`,_

 2V}^ E4_1  @

2VA^B^/ Û§°0W_,,,

3 X ª§°± 3 nW_²⁰_,,Ս,3 Ü. Ü. «

²Ö,Ս×,²+g,Õ+g

3 X ª§°± 3 nW_±,n⁰ 3 Ü. Ü. «

²+¦Õ+Ö d.

2.3.9

Relation to the disorder parameter Ý# of Li and Nattermann

M. S. Li and T. Nattermann³ defined the dimensionless parameter ζ# to control the relative disorder strength, and expand the random Gaussian disorder in renormalized Hermite polynomials to express the disorder term of GL free energy equation. In order to use the disorder parameter ζ# in our simulation, I calculated the relation equation of standard deviation Ú and disorder parameter ζ#.The disorder term of M. S. Li and T.

Nattermann is

( )^t 4N_t||^, 2.3.10

here N_t is real and Gaussian distributed with

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N_t

ÌÌÌÌÌÌÌÌÌ 0,

NÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌÌ_tN_t5  Þ^_^H^N_Ft  t5. 2.3.11

The typical fluctuations N_t ß ζ_ of the mean field transition temperature. They defined the disorder parameter with following relation

ށ  Þ_x/+⁹_hÏh9^/+ . 2.3.12

Connecting our disorder term with their notation, we have

Í^B _<x9^Š+J_:⁺_;Þ^_^H^, 2.3.13

use the reduce temperature W_; we can rewrite Í as follows

Í^B _àx^Š+ W_;^ށ^, 2.3.14

and substitute Eq.( 2.3.14) into Eq.( 2.3.8)

v_x9^ŠJ;KhÏ_:;/+w^Ú^ _à<x^Š+_+z+W_;^ށ^. 2.3.15

Thus, we have the relation between standard deviation Ú and disorder parameter ζ#. By controlling the specific ζ# to generate the corresponding complex random fields, we can get various degree disorder vortex systems.

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We can use Eq.(2.4.2) to calculated the thermodynamic quantities. For a example, the average of energy is

ëâì ∑_{ƒŠŠ σÏ}â?^hhâQã;

á .

2.4.3

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Chapter 3 Monte Carlo Method

3.1 Metropolis algorithm

The standard Monte Carlo method with Metropolis algorithm²¹ was used to simulate the two-dimensional pure and disordered vortex system. In the classic Metropolis method, we use a transition probability which depends on the difference of energy ∆! between the initial and trial configuration to determine whether the trial configuration is accepted or not.

Now I introduce my Monte Carlo method as follows. First, we choose an initial configuration and calculate the initial energy !_. Second, we choose a site P_íî ï^ðñ randomly and generate the trial configuration with P_í^òó by using the rule: P_í^òó ‰ P_í^ôŠ¥3 õ∆P, where ∆P is a complex number which is chosen randomly from the region

|Í?∆P| ö 1 and |Ø>∆P| ö 1 in the complex plane. Third, we calculate the energy !_ò of trial configuration and the difference of energy ∆!, here ∆!  !_ò !_. If ∆! ö 0, the system accepted the trial configuration, but if ∆! ³ 0, the trial configuration is accepted with a probability expù∆!. Generating a random number t uniformly in the

interval  0 , 1 ’, if t ö ?Yù∆! the trial configuration is accepted, otherwise it is rejected. This process is called Monte Carlo step/site (MCS/site). Note that the old configuration is still counted again for averaging if the trial configuration is rejected. By using Monte Carlo method, the system will fall into the stable states and reach the equilibration, and the characteristics of vortex system can be measured.

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3.2 Monte Carlo calculations

I used Eq.(3.2.1) to vary the value of a specific wave function coefficient P_í in our Monte Carlo simulation

P_Š^òó_,Š+  P_Š^ôŠ¥_,Š+ 3 δ_Šhsδ_Š+hs+∆, 3.2.1

here ∆ õ∆P. Note that í and © are vectors which composed of two reciprocal vectors )_ and )_. Furthermore we used Eq.(3.2.1) to calculate the energy of trial configuration and only discussed the changes of the summation of wave function coefficient product, the detail of Monte Carlo calculations are worked out in Appendix C. The summation of wave function coefficient product of trial configuration is

X P_Š^òó0_,Š+ P_Š^òó_,Š+

configuration, hence we can store it to simulate the vortex system more efficiently. The old calculation results always can be applied in new one and a lot of computer time is saved, the CPU time in one Monte Carlo step û }^. There are six different size (4 € 4, 6€ 6, 8 € 8, 10 € 10, 12 € 12, 16 € 16 numbers of vortices) systems in our simulation. We took 5 € 10^ü~1 € 10^à MC steps to reach the thermal equilibration and calculated the physical quantities over 1 € 10^à ~ 1 € 10^ý MC steps. The physical quantities were measured every

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30 ~ 50 MC steps. We control õ in a reasonable region to make the acceptance ratio is 0.3 ~ 0.4 and then the vortex system reach the thermal equilibrium state efficiently. All the simulations were started from the heating processes with the initial configuration which is defined as follows: P_/  I^|ƒ_Ž
^þ|

 , here P_/ is one of all coefficients of wave function and others are equal to zero, ù_ ~ 1.16 is the mean-field value of the Abrikosove ratio.

Chapter 4 Results for the clean system subject to thermal fluctuations

4.1 Abrikosov ratio, hexagonal symmetry

Abrikosov ratio explains the configuration of the vortex system. At low temperature, the system is in solid state and the vortices are arrayed regular as a atomic lattice, otherwise the vortices are arrayed as liquid in high temperature region. The definition of Abrikosov ratio is

ù_{  }^{ ë( |`|˜ ì}

vë( |`|_˜ ⁺ìw⁺. (4.1.1) The results of Abrikosov ratio for different size are shown in Fig. 4-1-1. The value of ù_ is close to the mean-field value 1.16 at low temperature and the vortices are close to each other. It explains that my Monte Carlo simulation is reasonable. The Monte Carlo data of various size system  are collapse onto a single curve unless W_; near the melting

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temperature. Note that the curves of the larger system size have slightly jumps while W_; ~ @_.

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4.2 Normalized specific heat

The definition of normalized specific heat is ^

 . Here

P_^¢ëv^M_;w^ì  ë^M_;ì^£, (4.2.1) which is the specific heat from the energy fluctuation and P is the mean-field value of the specific heat. The results of normalized specific heat for different size are shown in Fig.

4-2-1. The curve has a pick indicates that the vortex systems have a melting transition near the melting temperature. In high temperature region (above @_), the normalized specific heat decays quickly. temperature W_;. The squares, circles, triangles, inverted triangles, diamonds and pentangle correspond to system size   16, 36, 64, 100, 144, 256.

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4.3 Internal energy and its distribution

Internal energy for _   vortex system

The results of the dimensionless internal energy for   256 are shown in Fig.

4-3-1. I took 3 € 10^à Monte Carlo steps to measure this quantity, and used 3 € 10^à Monte Carlo steps for the equilibration. My Monte Carlo data are very close to the results which were obtained by Kato and Nagaosa and then the simulation method which I used is reasonable. Note that I didn’t find the indication of phase transition in the results. In order to discuss the phase transition of 2D vortex system, I measured the probability of energy distribution ! and the history of normalized internal energy.

-20 -15 -10 -5 0 5

-100 -80 -60 -40 -20 0

dimensionless internal energy <H> / T

a_T Ns 256

Fig. 4-3-1.

The dimensionless internal energy ^ëMì

; versus reduce temperature _W; for vortex number ₂₅₆.

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The probability of energy distribution n

The probability of energy distribution ! for various system sizes are shown in Fig. 4-3-2. As the result, ! has a double-peak, the right peak corresponds to higher energy which represents the liquid phase and left peak corresponds to lower energy which represents the solid phase. The double-peak distribution indicates that the vortex system has a first order phase transition when W_; ~ @_ (here W_;  12.5 for   100,

W_;  12.8 for   144, W_;  13.02 for   256).

-0.980 -0.975 -0.970 -0.965 -0.960 -0.955 -0.950 -0.945 -0.940 -0.01

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

P(E)

E

Ns 100 aT -12.5

(a)

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The energy distribution _! versus E with different vortex number:

(a)   100, (b)   144, (c)   256. Each temperature

approach the melting temperature of different Ns and the double peak is suggests that the first order phase transition exists in the melting process of the vortex lattice.

(c) (b)

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The history of the internal energy

Fig. 4-3-3 shows the history of the normalized internal energy of   256 vortex system at W_;  13.02, we applied the solid initial condition to simulate the vortices system and recorded the internal energy every Monte Carlo step. Note that we used

2 € 10^ü Monte Carlo steps to reach the equilibrium which are not shown in the result. The relaxation processes consists two regions and a sharp transition region. The lower internal energy region corresponds to the metastable state while the higher internal energy

corresponds to the stable one. Because of the sharp transition region is between the other two regions, the first order phase transition actually exists in the two locally stable states.

0 5000000 10000000 15000000 20000000

-0.975 -0.970 -0.965 -0.960 -0.955

normalized internal energy

Monte Carlo step Ns 256

a_T -13.02

Fig. 4-3-3.

The history of normalized internal energy E with _{256, W;} 

13.02. The sharp transition exists between the two regions with different energy. This phenomenon suggests the first order phase transition exists in the melting process.

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4. 4 Vortices configuration and Structure factor

Vortices configuration

In Fig. 4-4-1, we shown the snapshots of the spatial distribution of the magnitude of the order-parameter field |¨, |^ for W_; ³ @_ and W_; @_. There are 256

vortices in each system and we used 1 € 10^à Monte Carlo steps to reach the equilibration.

I plotted the snapshots in the form of a rectangle. The range of x and y is 0~}_{`</sub> and 0~}<sub>_</sub>, respectively, here }<sub>`} 3^/<V^/} and }_{_}  2V^/}/3^/<. As the results, (a) the vortices are arrayed regular like a lattice at W_;  15 @_, otherwise (b) they arrayed randomly at W_;  8 ³ @_.

(a)

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We transfer the spatial distribution into the momentum space by Fourier transform.

The Fourier transform of the density-density correlation function is shown as follows _‘  ( ( ë|¨t|_{” ”:} ^|¨t^B|^ì?^h/‘6h:8. 4.4.1

We transformed Eq.( 4.4.1) into the other form

_‘  4V^}^<?^h‘+/ë|∆‘|^ì, 4.4.2

with

Fig. 4-4-1

The snapshots of ₂₅₆ vortex position for (a) liquid state at W_;  8 (b) solid state at W_;  15. The range of X and Y is _0~}` and _{0~}_} respectively and the dark spots correspond to vortex cores.

(b)

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∆‘  X ?Y Z[V \ ^B2E_ E_

©

31

2 Y^3 2E_ ^BB2Y_3 _  Y_3 _’ad ?Y ¢[V 2  ^B

 ^B£ P_²⁰_{¦Š’,²+¦Š+’}P_©.

4.4.3

Here ‘  ± 3 n is a reciprocal lattice vector of the Abrokosov lattice and we replaced ‘E by ‘.

The results of the snapshots of the diffraction pattern are shown for W_; ³ @_ and W_; @_ in Fig. 4-4-2, respectively. I used ë|∆‘|^ì to characterize the Bragg peaks:

At W_;  15, the Bragg peaks with hexagonal symmetry are separated and very sharp, indicating the existence of the triangular lattice of the vortices. But the Bragg peaks are disappeared and the diagram has a circular shape at W_;  8. Thus, the most distinct difference of Bragg peaks between liquid phase and that of solid phase is the behavior of rotation symmetry. If the vortex system is in liquid state, the picture has rotation symmetry.

However, the rotation symmetry will be broken while the vortex system is in solid state.

Next, we calculated structure factor to obtain the melting temperature of the flux-line-lattice.

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Fig. 4-4-2.

The snapshots of the diffraction pattern _ë|∆‘|^_ì for (a) liquid state at _{W;  8} (b) solid state at _{W;  15}. Here we set ë|∆0|^ì is neglected.

(b) (a)

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Structure factor

The definition of structure factor is

S‘  _‘/= , 4.4.4

= is the area of the sample. In order to discuss the phase transition between the vortex liquid and vortex lattice, I calculated the structure factor S‘ at the maximum

position _{_}  4V^//3^/<.

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The temperature dependence of the structure factor is shown in Fig. 4-4-3. As the Monte Carlo data shows, the curves separate for W_; 8 and collapse onto a single curve for W_; ³ 8. The curves splay out at W_; 12 ˆ 2 where indicates the transition temperature of the flux line lattice.

Furthermore, Fig. 4-4-4 shows that the size dependence of the structure. While the vortex system is in quasisoild state for W_; @_ , the structure factors S‘ are proportion to the system size  . However, the Monte Carlo data of structure factor S‘ are almost constant while the vortex system is in quasi-liquid state for W_; ³ @_. As the result, our results are close to the results of Kato and Nagaosa¹, the slope of the dotted line is

approximately 0.97 and S‘ û , which corresponds to long-range order. The slope of

10 100 reduced temperature W_;. The squares, circles, triangles, inverted triangles, diamonds and pentangles correspond to system size W_;  18, 15, 13, 10, 7, 5. Here _S‘ is in units^;Š

9B.

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the broken line is approximately 0.85 and ‘ û ^ü/à, which corresponds to the size dependence at the melting point of the KTBHNY theory^14~17.

The melting temperature for infinite vortex system size

Now we turn back to discuss the melting temperature for infinite vortex system size.

I provided four different system sizes,   16^, 12^, 10^, 8^ and the results of the size dependence of the melting temperature @_ are shown in Fig. 4-4-5. For   16^, 12^, 10^, we find each double-peak of the internal energy distribution ! are almost the same high. For smaller system size   8^, I didn’t find the double-peak in the energy distribution diagram. I determined the melting temperature of this system by at what temperature the rotation invariance was disappeared. In Fig. 3-1-10, I fitted the data by the

0.000 0.025 0.050 0.075 0.100 0.125 0.150

The dependence of melting temperature _@ on system size

^h/. By fitting the Monte Carlo data linearly, we found the melting temperature _@  14.10932 for the infinite system size.

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linear equation and found the melting temperature @_  14.1 ˆ 0.1 for the infinite system size. Comparing with previous studies, @_  14.3 ˆ 0.2 for ~ ∞ by Kato and Nagaosa¹; @_  13.1 for   144 by Li and Nattermann³; @_  13.2 for

  12 € 14 by Hu and MacDonald². Thus, the melting temperature which I provided is close to the result of the previous researchers.

Chapter5 Quenched disorder

5.1 Comparison of the structure factor with that of the pure system

Fig. 5-1-1.

The wave vector dependence of the structure factor for disordered system at _{W;  15}. The squares, circles and triangles correspond to _{ζ#  0} (clean system), _{ζ#  0.03} and _{ζ#  0.2}, respectively.

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In this section, I added the disorder term to the GL free energy and discussed the difference between the clean and the disorder system. The results of ^_{_} dependence of structure factor for three different ζ# with   100, W_;  15 are shown in Fig. 5-1-1.

We took 1 € 10^à Monte Carlo steps to reach equilibrium and 2 € 10^à Monte Carlo steps for measure the quantity. As the result, the Bragg peaks are very sharp for ζ#  0 case but they become shorter for ζ#  0.03 case and ζ#  0.2 case. Obviously, we do not find the sharp peaks for ζ#  0.2 case. Next I discussed the snapshots of the spatial distribution of vortices position with different ζ# , the results are shown in Fig. 5-1-2. For (a) ζ#  0 case, the system has no disorder and the configuration of vortices similar to an Abrikosov lattice.

Then I applied weak disorder with (b) ζ#  0.03 to the system and found the vortices location became slightly irregular. The configuration of vortices is more irregular as the result of (c) ζ#  0.2 case. Thus, if we increase the value of disorder parameter ζ# , the spatial distribution of vortices position becomes more irregular.

(a)

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(c) Fig. 5-1-2.

The snapshots of the vortex position for (a) ζ#  0 (clean system), (b) ζ#  0.03, (c) ζ#  0.2 with   100, W_;  15. The range of x and y is _0~}` and _{0~}_} respectively and the dark spots correspond to vortex cores.

(b)

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I also measured the reduced temperature dependence of structure factor for various vortex system with weak disorder ζ#  0.01. There are six different system sizes in Fig.

5-1-3, each Monte Carlo data had been run 1 € 10^à Monte Carlo steps for equilibration and 2 € 10^à Monte Carlo steps for measuring the quantity. Besides, the average value was done by 80 disorder samples. As the results, the curves splay out at W_; 12 ˆ 2 and collapse onto a single curve for W_; ³ 7. I found the vortex system with weak disorder still has the melting transition between solid state and liquid state. It is reasonable that the melting temperature of the disorder system and that of pure system which we presented are similar since the disorder term I applied here is weak.

-20 -15 -10 -5 0

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5.2 The glass line of the disorder system

In this section, I tried to find the glass line of the disorder vortex system. I introduce another disorder parameter t and glass line temperature W_;^ of the theories of

B. Rosenstein and D. Li¹³. The definitions are

t _<x^hÏ_+ÏF+⁺_√/^ÙB , (5.2.1)

W_;^  2√2 -√t _√”^2, (5.2.2)

[ is the Ginzburg number. In theory, the vortex glass become vortex liquid while W_; ³ W_;^, see Fig. 5-2-1. If the disorder parameter t is fixed, we can obtain the corresponding value W_;^  Ü. Hence, I simulated the disorder vortex system with fixed t and tried to find the theoretically value of W_;^.

Fig. 5-2-1.

The phase diagram of vortex glass and vortex liquid which are separated by _W;^ curve

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The magnetization

The vortex systems were simulated with the disorder parameter t  0.32. The Monte Carlo data had been run 1 € 10^à Monte Carlo steps for equilibration and 2 € 10^à Monte Carlo steps for measuring the quantity. The statistics and averages were done by 80 disorder samples. Fig. 5-2-2 shows the system size dependence of magnetization for

various W_; (5, 6, 7, 8, 9, 10). As the results, the disorder average of the magnetization converges very fast with the system size. The physical quantities don’t depend on the system size, so that my calculation is meaningful.

20 40 60 80 100 120 140 160

The system size dependence of magnetization for various _W;. The squares, circles, triangles, inverted triangles, diamonds and pentangles correspond to system size _W;  5, 6, 7, 8, 9, 10, respectively.

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The second moments

The results of the system size dependence of second moments for various W_; are shown in Fig. 5-2-3. As the result, the curves decay as the system size increases. In the theory, the second moments converge to a finite value in the vortex glass state; however, the second moments converge to zero in the vortex liquid state. Here I set the disorder parameter t  0.32 and W_;^~  8 theoretically. In Fig. 5-2-3, the second moments don’t seem to converge to a finite value in this range of the reduce temperature. Thus, I did not find the glass line in my Monte Carlo simulation results. The detail values of other moments and the magnetization for various W_; are shown in Table 5-2-1.

20 40 60 80 100 120 140 160

1 2

Second moments m2

Ns

a_T= -5 aT= -6 aT= -7 aT= -8 aT= -9 a_T= -10

Fig. 5-2-3.

The system size dependence of second moments for various _W;. The squares, circles, triangles, inverted triangles, diamonds and pentangles correspond to system size _W;  5, 6, 7, 8, 9, 10, respectively.

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Table 5-2-1

The Monte Carlo data of the nth moments and magnetization for various reduced temperature W_;.

Ns=36 2th 4th 6th ë( |¨t|ÌÌÌÌÌÌÌÌÌÌÌÌÌ_” ^ì /

-5.000000 1.903627 0.968647 0.785916 33.101730

-6.000000 1.496095 0.981333 0.892610 36.307093

-7.000000 1.465578 1.162532 1.412543 40.434934

-8.000000 1.089013 0.759833 0.472182 45.165741

-9.000000 0.668829 0.850774 0.602358 49.849915

-10.000000 0.369992 1.161049 1.184144 54.602529

Ns=64 2th 4th 6th ë( |¨t|Ì_”ÌÌÌÌÌÌÌÌÌÌÌÌ^ì /

-5.000000 1.181913 1.720579 3.088818 33.150994

-6.000000 0.985165 0.801090 0.487614 36.375821

-7.000000 0.888345 0.895439 0.695364 40.407829

-8.000000 0.363408 1.167023 1.046770 45.035289

-9.000000 0.409480 0.766420 0.508653 49.753961

-10.000000 0.514633 1.354023 1.499346 54.862039

Ns=100 2th 4th 6th ë( |¨t|ÌÌÌÌÌÌÌÌÌÌÌÌÌ_” ^ì /

-5.000000 0.680257 0.942989 0.748037 33.006740

-6.000000 0.523575 0.804231 0.579638 36.496617

-7.000000 0.522493 0.745477 0.418757 40.465406

-8.000000 0.249057 1.048304 0.979836 44.953665

-9.000000 0.215101 0.875568 0.593321 49.882716

-10.000000 0.237399 0.796513 0.490476 54.705229

Ns=144 2th 4th 6th ë( |¨t|ÌÌÌÌÌÌÌÌÌÌÌÌÌ_” ^ì /

-5.000000 0.590747 0.921689 0.832652 33.065012

-6.000000 0.437729 1.067987 1.070969 36.591659

-7.000000 0.387085 1.045748 0.974747 40.675267

-8.000000 0.283228 0.772528 0.522839 45.222984

-9.000000 0.159739 1.181599 1.227355 50.131457

-10.000000 0.167486 1.089205 1.051271 54.852637

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Chapter 6 Conclusion

I have studied in this thesis certain properties of idealized 2D type II superconductor

I have studied in this thesis certain properties of idealized 2D type II superconductor

在文檔中 二維Ginzburg-Landau模型模擬第二類超導的熱擾動和無序現象 (頁 15-0)