Glass transition and the replica symmetry breaking in vortex matter: MC study

Computer Physics Communications 182 (2011) 55–57

Contents lists available atScienceDirect

Computer Physics Communications

www.elsevier.com/locate/cpc

Glass transition and the replica symmetry breaking in vortex matter: MC study

Dingping Li

a

, B. Rosenstein

b

,

∗

, Shou Chen

a

, H.H. Lin

b

, D. Berco

b

a_{Department of Physics, Peking University, Beijing, 100871, China}

b_{Electrophysics Department, National Chiao Tung University, Hsinchu 30050, Taiwan, ROC}

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 1 March 2010

Received in revised form 20 July 2010 Accepted 28 July 2010

Available online 10 August 2010 Keywords:

Vortex matter Glass phase Melting

We investigate effects of disorder and thermal fluctuations on the Abrikosov state of type II superconduc-tors applying the Monte Carlo method to Ginzburg–Landau theory to confirm earlier replica calculation. The vortex phase diagram has two transition lines, the melting line and the vortex glass transition lines in both crystalline and homogeneous states. The glass line is always a continuous transition, while the melting line is the first order.

©2010 Elsevier B.V. All rights reserved.

Combination of disorder, interactions and thermal fluctuation in condensed matter systems often results in a complicated phase di-agram containing a variety of glassy states which are notoriously difficult to describe theoretically. Systems of Abrikosov vortices created by magnetic field in type II superconductors offer a unique testing ground for experimental verification of theoretical meth-ods attempting to describe these complex states. In this system all three ingredients are naturally present. Interaction between vor-tices is quite strong and at zero temperature and without disorder creates highly correlated systems like Abrikosov lattice and impu-rities are present naturally. In highly anisotropic (quasi 2D) high Tc superconductors like BSCCO (BSCCO) the Ginzburg number Gi

characterizing the strength of thermal fluctuations is not small and thereby thermal fluctuations are strong. They compete with inter-actions leading to the lattice melting into a homogeneous vortex liquid state and also effectively reducing disorder leading to ther-mal depinning of vortices and demise of the glassy state in large portions of the H –T phase diagram. The BSCCO phase diagram ob-tained recently using the magnetization method revealed four dis-tinct phases: Bragg glass (BG), vortex glass (VG), liquid (L) and the Abrikosov lattice (A)[1]. The last phase which exists at elevated temperatures, weaker disorder and relatively strong interactions is interesting, since theoretically there is no clear agreement even on its existence[2,3,7].

Since in most relevant experiments on strongly type II (

κ

=

λ/ξ



1

λ

-penetration depth,

ξ

-coherence length) high Tc

su-perconductors fields are much larger than the lower critical field Hc1(T

)

, magnetic fields of the vortices overlap and induction B becomes essentially homogeneous. Instead of flux lines one should

*

Corresponding author.

E-mail address:[email protected](B. Rosenstein).

return to a less phenomenological Ginzburg–Landau model in con-stant field.

Numerical simulation of the X Y model generally have not de-tected the Abrikosov lattice phase, [4], although very recent nu-merical dynamic simulation is consistent with the four phase di-agram with the Abrikosov lattice (A) phase called “floating solid” phase. However it is interpreted as a finite size effect [5]. Disor-dered GL model was simulated only in 2D and within the lowest Landau level (LLL) approximation discussed below[6], exhibit the first order homogeneous–crystalline transition, but no clear evi-dence of the glassy behavior was demonstrated. Analytical meth-ods include the replica [2,3] and dynamic approach [8] and are summarized below.

In this note we study the disordered 2D Ginzburg–Landau model within the lowest Landau level (LLL) approximation numer-ically using the Monte Carlo (MC) simulation. Our starting point is the Gibbs energy:

G

=

Lz



d2r h

¯

2 2m∗

|

D

ψ

|

2

₊

_α

_T c

(

1

−

t

)

×



1

+

W

(

r

)



|ψ|

2

+

β

2

|ψ|

4

_,

₍₁₎

where covariant derivative is defined by D

≡ ∇ −

i_¯hcΦe∗

0A, with A being the vector potential in Landau gauge A

= (

B y

,

0

)

, t

≡

T

/

Tc

and Lz is the thickness. Mesoscopic thermal fluctuations are

ac-counted for via statistical sum Z

=



_ψexp

{−

G

[ψ

∗

, ψ

]/

T

}

. Point-like (

δ

Tc) quenched disorder on the mesoscopic scale is described

by the random potential with variance W

(

r

)

W

(

r

)

=

R

ξ

2

δ(

r

−

r

)

. In wide range of fields and temperatures, see Ref. [9]for details, the model can be simplified by retaining just the lowest Landau level. Unit of magnetic field will be Hc2(T

=

0

)

so that b

=

B

/

Hc2 0010-4655/$ – see front matter ©2010 Elsevier B.V. All rights reserved.

56 D. Li et al. / Computer Physics Communications 182 (2011) 55–57

Fig. 1. Generic phase diagram of the vortex matter. (a) The order–disorder line (red) separates the crystalline phases (A and BG) from the homogeneous phases (L and VG).

The glass transition line is the blue line in homogeneous phase and the light green line in the crystalline phase. (b) is an enlarged region near the crossing point. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

and unit of length—magnetic length lH

= ξ/

b1/2. We will use the

quasimomentum basis of Ref. [9], that has an advantage for the MC simulation. Our sample has he following dimensions: Ld1

,

Ld2

(area is 2

π

L2 containing N

=

L2 vortices), where Abrikosov lattice vectors are d1

= (

a

,

0

),

d2

=

a

(

1₂

,

√ 3 2

)

with a

=



4

π

/

√

3. Periodic boundary conditions make quasimomentum discrete taking values

k

=

n1

L



d1

+

n2

L



d2, ni

=

0

, . . . ,

L

−

1.

The Boltzmann weight of the reduced (LLL) model is

g

=

1 4

π



x,y aT



ϕ

(

x

,

y

)



2

+

w

(

r

)



ϕ

(

x

,

y

)



2

+

1 2



ϕ

(

x

,

y

)



4

,

w

(

x

,

y

)

w

x

,

y



=

4

π

R

δ

x

−

x



δ

y

−

y



,

(2) where the 2D LLL reduced temperature aT

≡ −

H_κc2



Φ0Lz 4πH T

(

1

−

t

−

h

)

, and R

=

(1₄−_ωt)_t2R, w

(

x

,

y

)

= −



π ωbt

(

1

−

t

)

W

(

r

)

where

ω

=

m∗β

2_¯h2_αLz. Our simulation has no spatial grid avoiding the problem of the artificial pinning by the grid[4,5]. We use Metropolis algorithm to simulate up to 100 different disorder configurations for sizes up to N

=

16

×

16. Thermalization was achieved by 4

×

105–4

×

106 sweeps and results were obtained from runs up to 107 _sweeps.

Clean system was simulated with results similar to those obtained in other simulations and is consistent with theory presented [3] and plotted in [1]. The generic phase diagram is given in Fig. 1.

Homogeneous phases (L and VG) appear at larger scaled temper-ature aT than the crystalline ones (A, BG), while pinned (glassy)

phases (VG, BG) appear at larger disorder R than the unpinned ones (L, A). In particularFig. 1(b) shows that there is a tiny split in the glass line into two tricritical points, see below for explanation. In Fig. 2 points in the

(

aT

,

R

)

parameter space in which the

MC simulations were performed. We fix R

=

0

.

001, aT is varied

from

−

15

,

−

16

,

17

,

18

,

19

,

20. Structure functions are presented in Fig. 2. One clearly observes that AL with clear Bragg peaks be-comes a Bragg lattice with diffuse peaks and become homoge-neous glassy state at large disorder.

The glass line in the homogeneous phase was calculated using the replica method in[3]and using the dynamic approach in[8]:

a_Tg

=

4R

√

−

1

2R

.

(3)

The glass transition in the crystalline phase follows a more com-plicated formula;

a_Tg

= −

e−1

−

Re

/

2

−

4

(

2R

)

−1/2

+

3

(

2R

)

1/2

,

(4) where e is a solution of cubic equation,

−

R2e3

+

Re3

− (

2R

)

1/2e2

+ (

2R

)

3/2e2

D. Li et al. / Computer Physics Communications 182 (2011) 55–57 57

Fig. 2. Structure functions with disorder at different temperature. (a)–(c) represent

the structure functions at aT=15,16,17 respectively and they have sharp Bragg peaks and they belong to Abrikosov states. (d), (e) have smeared Bragg peak and they belong to Bragg glass. (f) restores rotational symmetry and it belongs to vortex glass.

At small R, eO

= (

2R

)

−1/2

+ · · ·

and the correction to the line is

of order R3/2 _{[1], a}g T

=

4

R−1

√

2R

+

O

(

R

3/2

₎

_{. There are two tricritical}

points separated only slightly (since at that point R is small), so

that the crystalline glass line is at higher temperatures compared to the homogeneous one.

To summarize, the MC simulation confirms the replica results for the four phase picture of vortex matter phase diagram in 2D far from the Hc1(T

)

line. The glassy phases in both the homogeneous and the crystalline segments are established, although it is difficult at this stage to confirm details of the phase boundaries.

Acknowledgements

It is a pleasure to thank E. Zeldov, V. Vinokur, X. Hu, B. Shieh, R.L. Hung for illuminating discussions. B.R. is grateful to Univer-sity Center of Samaria and Bar Ilan UniverUniver-sity for hospitality dur-ing sabbatical leave, D.L. is grateful to National Chiao Tung Uni-versity for hospitality. Work supported by NSC of R.O.C. grant, NSC#982112M009048 and MOE ATU program, and National Sci-ence Foundation of China Grants Nos. 90403002 and 10974001.

References

[1] H. Biedenkopf, et al., Phys. Rev. Lett. 95 (2005) 257004; H. Biedenkopf, et al., Phys. Rev. Lett. 98 (2007) 167004. [2] D. Li, B. Rosenstein, Phys. Rev. Lett. 90 (2003) 167004;

B. Rosenstein, D. Li, Rev. Mod. Phys. 82 (2010) 109.

[3] D. Li, B. Rosenstein, V. Vinokur, J. Supercon. Novel Magnet. 19 (2006) 369. [4] Y. Nonomura, X. Hu, Phys. Rev. Lett. 28 (2001) 5140.

[5] P. Olsson, S. Teitel, Phys. Rev. B 79 (2009) 214503. [6] M. Li, T. Nattermann, Phys. Rev. B 67 (2003) 184520. [7] T. Giamarchi, P. Le Doussal, Phys. Rev. B 52 (1995) 1242. [8] A.T. Dorsey, M. Huang, M.P.A. Fisher, Phys. Rev. B 45 (1992) 523;

B. Rosenstein, V. Zhuravlev, Phys. Rev. B 76 (2007) 014507. [9] D. Li, B. Rosenstein, Phys. Rev. B 65 (2001) 024514.