Download Effective-field theory on the transverse Ising model under a time

Physics Letters A 374 (2010) 1885–1888
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Physics Letters A
www.elsevier.com/locate/pla
Effective-field theory on the transverse Ising model under
a time oscillating longitudinal field
Xiaoling Shi a , Guozhu Wei b,c,∗
a
b
c
College of Sciences, Liaoning University of Petroleum and Chemical Technology, Fushun 113001, China
College of Sciences, Northeastern University, Shenyang 110004, China
International Center for Material Physics, Academia Sinica, Shenyang 110015, China
a r t i c l e
i n f o
Article history:
Received 10 December 2009
Received in revised form 28 January 2010
Accepted 17 February 2010
Available online 21 February 2010
Communicated by A.R. Bishop
Keywords:
Transverse Ising model
Effective-field theory
Dynamic phase transition
a b s t r a c t
As an analytical method, the effective-field theory (EFT) is used to study an Ising spin system in a
transverse magnetic field under a time oscillating longitudinal field. The effective-field equations of
motion of the average magnetization are given for the square lattice (Z = 4). In the longitudinal field
amplitude h0 / Z J -transverse field Γ / Z J plane, the phase boundary separating the dynamic ordered
and the disordered phase also has been drawn, and there is no dynamical tricritical point on the
dynamic phase transition boundary. The dependence of the critical temperature on the transverse field is
calculated and phase diagrams are presented. We also make the compare results of EFT with that given
by using the mean field theory (MFT).
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
In recent years there have been many theoretical studies on
cooperative spin systems subject to external magnetic fields dependent on time [1–18]. These models show the nonequilibrium
dynamical phase transition and have been arousing great interest
for their intriguing physics. Some aspects of such a nonequilibrium dynamical phase transition have been observed experimentally [19–23].
In order to study the effect of quantum fluctuations in classical
spin models, Ising model in a transverse field has been investigated extensively [24–42]. The transverse Ising model is used to
represent the order-disorder transition in hydrogen-bonded ferroelectrics and the possibility of tuning the asymmetry of the
double-well and the transverse field by changing the external pressure on the hydrogen-bonded ferroelectrics [43]. The transverse
Ising model, due to its connection to a wide variety of systems
found in condensed matter physics, has been used to understand
the response of the quantum spin systems to oscillating longitudinal fields. The critical and magnetic properties of the transverse
Ising model under a time oscillating longitudinal fields have also
been studied by using the mean-field approximation [6]. But we
know that the mean-field theory neglects nontrivial thermal fluctuations to investigate this nonequilibrium dynamic transition, the
*
Corresponding author at: College of Sciences, Northeastern University, Shenyang
110004, China.
E-mail address: [email protected] (G. Wei).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.physleta.2010.02.049
results given by MFT need further investigated by using more reliable techniques. Santos et al. studied ground-state properties of
this kinetic quantum model by the mean-field theory and the
Monte Carlo simulation [28]. But up to now few people, as far as
we know, have ever touched upon this system within the effectivefield theory with correlations. The correlated effective-field theory
(EFT) was first proposed by Kaneyoshi et al. [44] and has been
successfully applied to some equilibrium Ising models without the
oscillating magnetic field [34–47] and nonequilibrium Ising models under the oscillating magnetic field [48,49]. The EFT considers
partially the spin–spin correlations and results in an improvement
over the mean-field theory. In this Letter we introduce the correlated effective-field theory, as an analytical method, to study the
transverse Ising model under a time oscillating field, and compare
the results given by EFT with the results given by MFT.
The layout of this Letter is as follows. In Section 2, we briefly
present the EFT method we used. The results and discussion are
presented in Section 3. In Section 4, we summarize our conclusions.
2. Formulation
We consider a ferromagnetically interacting Ising model placed
in two types of external fields: a transverse field of constant magnitude and a longitudinal field which depends on time. The general
system can thus be described by the Hamiltonian given by
H=−
i , j J i j σiz σ jz − h(t )
i
σiz − Γ
i
σix .
(1)
1886
X. Shi, G. Wei / Physics Letters A 374 (2010) 1885–1888
Here σix and σiz are the Pauli matrices, J i j represents the spin–
spin interaction strength between sites i and j, the sun Σi is
carried out over all the sites, the sun Σi , j is carried out over all
the distinct nearest-neighbor pairs, and Γ represents the strength
of the transverse field, and h(t ) is a time-dependent longitudinal
field given by h(t ) = h0 sin(ωt ). The system is in contact with an
isothermal heat bath at temperature T . For simplicity all J i j are
taken equal to a constant J > 0.
In order to obtain the averaged magnetization within the EFT,
we write the Hamiltonian in the following form:
H = Hi + H ,
(2)
where
Hi = −σiz ( E i + h) − σix Γ,
Ei = J
σ jz ,
j
H = − J
σiz σ jz − h
i =i , j (3a)
(3b)
σiz − Γ
i
σix .
(3c)
i
Here, Hi includes all parts of H associated with the site i, E i is
the effective field of site i, and H represents the rest of the H
and dose not depend on the site i. We note that for “classical systems” Hi and H commute, while according to the effective-field
theory, for the transverse Ising model, Hi and H do not commute.
Following the original work of Sa Barreto et al. [34,35], we can neglect the fact that Hi and H do not commute. This approximate
relation has been accepted as a reasonable starting point in many
studies of the transverse Ising model, and is the method that was
adopted in Refs. [36–42]. In the limiting Γ → 0, the Hamiltonian
contains only σiz and the relationship is then exact. By introducing the differential operator technique and the Van der Waerden
identity, we obtain the average magnetization
σiz = Ei + h
tanh β ( E i + h)2 + Γ 2
( E i + h)2 + Γ 2
= (a + bM z ) Z f (x)
x=0 ,
where β =
1
kB T
(4)
, Z = 4 is the coordination number, and
a = cosh( J ∇),
b = sinh( J ∇),
x+h
F (x) = tanh β (x + h)2 + Γ 2 .
(x + h)2 + Γ 2
(5)
The equation for the dynamics of the magnetization of a magnet in the constant transverse field and the oscillating longitudinal
field can be written in the effective-field approximation as [48]
dM z
dt
= − M z + a0 + a1 M z + a2 M 2z + a3 M 3z + a4 M 4z .
(6)
The coefficients ai (i = 0–4) can be easily calculated employing
a mathematical relation exp(a∇) f (x) = f (x + a) (see Appendix A).
And the temperature T , the longitudinal field h and the transverse
field Γ are measure in units of Z J . Eq. (6) can be solved by using
the fourth-order Runge–Kutta method.
By defining the dynamic order parameter as the time-averaged
magnetization over a period of the oscillating magnetic field [1]
ω
Qz =
2π
M z (t ) dt
(7)
the two types of solutions can be identified: a symmetric one
where M z (t ) follows the longitudinal field oscillating around zero
giving Q z = 0, and an antisymmetric one where M z (t ) does not
follow the longitudinal field and oscillates around a finite value
different from zero, such that Q z = 0.
Fig. 1. Phase diagram of the dynamic phase boundary in the h0 / Z J –Γ / Z J plane for
T / Z J = 0.2. The dashed line is the boundary of the first-order transition and the
solid line is the boundary of the second-order transition, and the number accompanying each curve is the value of ω .
3. Results and discussion
Some typical results of the finite temperature dynamic phase
transition diagram in h0 / Z J –Γ / Z J plane are depicted in Fig. 1 for
T / Z J = 0.2. In the phase diagrams, the solid and dashed lines indicate, respectively, the second order and the first order dynamic
phase transition. In Fig. 1, we can see that the dynamic phase
diagram comprises a paramagnetic phase ( Q z = 0) at high value
of the longitudinal field amplitude h0 / Z J and a ferromagnetic
phase ( Q z = 0) at low value of the longitudinal field amplitude
h0 / Z J for a fixed value of the transverse field Γ / Z J . In the region
ω < 0.2292, the phase transition is always of first order. In the region ω > 0.2292, the phase transition is always of second order.
There is no tricritical point on the phase transition line. Compared
with the MFT result which claimed there is a tricritical point on
the phase boundary line, we know that the EFT results introduce
thermal fluctuations to obtain the correct result while the MFT totally neglects thermal fluctuations. And these finite temperature
results given by the EFT coincide with the results given by the MC
simulation at zero temperature [28] which also claimed there is no
tricritical point in the quantum Ising model.
Figs. 2(a) and 2(b) express the behavior of dynamic order parameter Q z as a function of transverse fields with longitudinal field
amplitude h0 / Z J = 0.28 and h0 / Z J = 0.125, respectively. From
Fig. 2(a), we can see that with the increase of the longitudinal field
frequency ω the dynamic order parameter first suddenly changes
to zero, which indicates the first-order dynamic phase transition and then continuously decreases to zero, which indicate the
second-order dynamic phase transition for fixed longitudinal fields
and temperature. And the critical transverse field of the dynamic
transition point increases with increasing the longitudinal field frequency ω for a given longitudinal field amplitude h0 / Z J > 0. This
fact is easily understood physically. As is well known, the dynamic
breaking of symmetry arises due to the competing time scales
of the oscillating field and the response magnetization. With increasing the longitudinal field frequency ω , the effective relaxation
lag of the response magnetization becomes large, thus the critical
transverse field of the dynamic transition point increases.
These values can be compared with the results given by meanfield theory in Ref. [6]. From Fig. 2(a), we find that the dynamic
order parameter is discontinuous obtained by EFT which indicates
the first-order dynamic phase transition, while that obtained by
X. Shi, G. Wei / Physics Letters A 374 (2010) 1885–1888
(a)
1887
(a)
Fig. 2. The transverse field variation of Q z / Z J for T / Z J = 0.2 and for fixed field
amplitude. (a) h0 / Z J = 0.28. (b) h0 / Z J = 0.125.
Fig. 4. The temperature variation of Q z / Z J for
plitude. (a) h0 / Z J = 0.28. (b) h0 / Z J = 0.1.
Fig. 3. Phase diagram of the dynamic phase boundary in the h0 / Z J –T / Z J plane for
ω = 0.0314. The dashed line is the boundary of the first-order transition and the
number accompanying each curve is the value of transverse field Γ / Z J .
MFT is continuous for ω = 0.0314 and h0 / Z J = 0.28. By comparing, we find that the critical transverse field of dynamic phase
transition point given by the EFT is lower than that of the MFT.
We know that the EFT considers partially the spin–spin correlations while the MET neglects the correlations of spin fluctuations.
These results given above indicate that the thermal fluctuations
play the major role in the dynamic phase transition. The results
given by the EFT for ω = 0.00628 may also be compared with the
ground-state phase diagram given by the Monte Carlo simulation
in Fig. 3 of Ref. [28]. For h0 / Z J = 0.125, Γc / Z J = 0.383 in EFT,
while Γc / Z J = 0.175 in MC simulation. We know that at finite
temperature, the critical value Γc / Z J of transverse field should be
lower than that of at zero temperature.
Some typical results of the dynamic phase transition diagram in
h0 / Z J –T / Z J plane are depicted in Fig. 3 for ω = 0.0314. In Fig. 3,
we can see that the phase diagram consists of entirely first order
transition lines, and the tricritical point does not exist. And it indicates that the dynamic phase diagram depends sensitively on the
transverse field. The smaller the transverse field Γ , the larger the
range of the ferromagnetic phase ( Q z = 0) in the dynamic phase
diagram.
In order to compare to the mean-field results and to see more
clearly the effect of the thermal fluctuations on the dynamic transition point, the temperature variations of the dynamic order parameter Q z for fixed field amplitude h0 / Z J = 0.28 and h0 / Z J =
ω = 0.0314 and for fixed field am-
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X. Shi, G. Wei / Physics Letters A 374 (2010) 1885–1888
0.1 are plotted in Fig. 4. From Fig. 4, we can see that the dynamic order parameter always changes to zero discontinuously.
Compared to the results given by mean-field theory in Fig. 9(a)
of Ref. [6], we find that the dynamic order parameter is discontinuous obtained by EFT, while that obtained by MFT is continuous
for ω = 0.0314 and h0 / Z J = 0.28. And the temperature of the dynamic phase transition point of the EFT is lower than that of the
MFT. Different results given by using the different theories indicate
that the correlations of spin fluctuations play an important role in
the dynamic phase transition.
4. Conclusion
The dynamical response of the transverse Ising model under
a sinusoidal oscillating longitudinal field has been studied by an
effective-field theory with correlations. We have found that the
system exhibits a continuous-phase transition line in the region
ω > 0.2292, while the system exhibits discontinuous-phase transition line in the region ω < 0.2292. But the system does not show
a dynamical tricritical point. The dynamic phase boundary is found
to be frequency of longitudinal field dependent, and the transverse field also has important effect on the magnetic properties
of kinetic Ising model. By comparing with the results given by using MFT for the given ω and h0 , the temperature of the dynamic
transition point or the critical transverse field is always lower. The
results given by EFT indicate that the thermal fluctuations play an
important role on the dynamic critical properties of the transverse
Ising model.
Appendix A
The coefficients ai (i = 0–4) in Eq. (6) can be easily calculated
employing a mathematical relation exp(a∇) f (x) = f (x + a) and are
given by
a0 = cosh4 ( J ∇) f (x)
x=0
=
1 16
f (4 J ) + 4 f (2 J ) + 6 f (0)
+ 4 f (−2 J ) + f (−4 J ) ,
a1 = 4 cosh3 ( J ∇) sinh( J ∇) f (x)
x=0
1
=
f (4 J ) + 2 f (2 J ) − 2 f (−2 J ) − f (−4 J ) ,
4
a2 = 6 cosh2 ( J ∇) sinh2 ( J ∇) f (x)
x=0
3
=
f (4 J ) − 2 f (0) + f (−4 J ) ,
8
a3 = 4 cosh( J ∇) sinh3 ( J ∇) f (x)
x=0
1
=
f (4 J ) − 2 f (2 J ) + 2 f (−2 J ) − f (−4 J ) ,
4
(A.1)
(A.2)
(A.3)
(A.4)
a4 = sinh4 ( J ∇) f (x)
x=0
=
1 f (4 J ) − 4 f (2 J ) + 6 f (0)
16
− 4 f (−2 J ) + f (−4 J ) .
(A.5)
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]
[46]
[47]
[48]
[49]
T. Tome, M.J. de Oliveira, Phys. Rev. A 41 (1990) 4251.
W.S. Lo, R.A. Pelcovits, Phys. Rev. A 41 (1990) 7471.
M.F. Zimmer, Phys. Rev. E 47 (1993) 3950.
C.N. Luse, A. Zangwill, Phys. Rev. E 50 (1994) 224.
M. Acharyya, B.K. Chakrabarti, in: Annu. Rev. Comput. Phys., vol. 1, 1994,
p. 107.
M. Acharyya, B.K. Chakrabarti, Phys. Rev. B 52 (1995) 6550.
M. Acharyya, Phys. Rev. E 56 (1997) 2407.
M. Acharyya, Phys. Rev. E 58 (1998) 179.
M. Acharyya, Phys. Rev. E 59 (1999) 218.
M. Acharyya, Phys. Rev. E 69 (2004) 027105.
S.W. Sides, P.A. Rikvold, M.A. Novotny, Phys. Rev. Lett. 81 (1998) 834.
S.W. Sides, P.A. Rikvold, M.A. Novotny, Phys. Rev. E 57 (1998) 6512.
S.W. Sides, P.A. Rikvold, M.A. Novotny, Phys. Rev. E 59 (1999) 2710.
H. Fujisaka, H. Tutu, P.A. Rikvold, Phys. Rev. E 63 (2001) 036109.
G. Korniss, P.A. Rikvold, M.A. Novotny, Phys. Rev. E 66 (2002) 056127.
H. Jang, M.J. Grimson, C.K. Hall, Phys. Rev. B 67 (2003) 094411.
Han Zhu, Shuai Dong, J.-M. Liu, Phys. Rev. B 70 (2004) 132403.
G. Berkolaiko, M. Grinfeld, Phys. Rev. E 76 (2007) 061110.
Y.L. He, G.C. Wang, Phys. Rev. Lett. 70 (1993) 2336.
Q. Jiang, H.-N. Yang, G.C. Wang, Phys. Rev. B 52 (1995) 14911.
W.Y. Lee, B.C. Choi, J. Lee, C.C. Yao, Y.B. Xu, D.G. Hasko, J.A.C. Bland, Appl. Phys.
Lett. 74 (1999) 1609.
W.Y. Lee, B.C. Choi, J. Lee, C.C. Yao, Y.B. Xu, D.G. Hasko, J.A.C. Bland, Phys. Rev.
B 60 (1999) 10216.
B.C. Choi, W.Y. Lee, A. Samad, J.A.C. Bland, Phys. Rev. B 60 (1999) 11906.
R.B. Stinchcombe, J. Phys. C: Solid State Phys. 6 (1973) 2459.
M. Acharyya, B.K. Chakrabarti, R.B. Stinchcombe, J. Phys. A 27 (1994) 1533.
V. Banerjee, S. Dattagupta, P. Sen, Phys. Rev. E 52 (1995) 1436.
M.J. de Oliveira, J.R.N. Chiappin, Physica A 238 (1997) 307.
M. Santos, M.J. de Oliveira, Int. J. Mod. Phys. B 13 (1999) 207.
M. Santos, Phys. Rev. E 61 (2000) 7204.
W. Jiang, G.Z. Wei, Z.H. Xin, Physica A 293 (2001) 455.
T. Kaneyoshi, Physica A 319 (2003) 355.
A.A. Ovcbinnikov, D.V. Dmitricv, V.Y. Krivnov, V.O. Cheranovskii, Phys. Rev. B 68
(2003) 214406.
B.H. Teng, H.K. Sy, Physica B 348 (2004) 485.
F.C. Sa Barreto, I.P. Fittipaldi, B. Zeks, Ferroelectrics 39 (1981) 1103.
F.C. Sa Barreto, I.P. Fittipaldi, Physica A 129 (1985) 323.
T. Kaneyoshi, E.F. Sarmento, I.P. Fittipaldi, Phys. Rev. B 38 (1988) 2649.
T. Kaneyoshi, M. Jascur, I.P. Fittipaldi, Phys. Rev. B 48 (1993) 250.
E.F. Sarmento, T. Kaneyoshi, Phys. Rev. B 48 (1993) 3232.
A. Elkouraychi, M. Saber, J.W. Tucker, Physica A 213 (1995) 576.
M. Saber, J.W. Tucker, Physica A 217 (1995) 407.
Shi-Lei Yan, Chuan-zhang Yang, Phys. Rev. B 57 (1998) 3512.
Wei Jiang, Guozhu Wei, Physica A 284 (2000) 215.
B.K. Chakrabarti, A. Dutta, P. Sen, Lect. Notes Phys. M 41 (1996).
T. Kaneyoshi, I.P. Fittipaldi, R. Honmura, T. Manabe, Phys. Rev. B 24 (1981) 481.
R. Honmura, Phys. Rev. B 30 (1984) 348.
E.F. Sarmento, T. Kaneyoshi, Phys. Rev. B 40 (1989) 2529.
A. Bobak, M. Jaščur, Phys. Rev. B 51 (1995) 11533.
Xiaoling Shi, Guozhu Wei, Lin Li, Phys. Lett. A 372 (2008) 5922.
Shi Xiaoling, Wei Guozhu, Commun. Theor. Phys. 51 (2009) 927.