Effect of Elastic Deformations on the Critical Behavior of Three-Dimensional Systems with Long-Range Interaction

0.1 MB·English

by  S. V. Belim

APS/123-QED
Effect of Elastic Deformations on the Critical Behavior of Three-Dimensional Systems
with Long-Range Interaction
S.V. Belim∗
Omsk State University, 55-a, pr. Mira, Omsk, Russia, 644077 \\
5 (Dated: February 2, 2008)
0
Afield-theoreticaldescriptionofthebehaviorofcompressibleIsingsystemswithlong-rangeinter-
0
actionsispresented. Thedescriptionisperformedinthetwo-loopapproximationinthreedimensions
2
withtheuseofthePadBorelresummationtechnique. Therenormalizationgroupequationsareana-

n lyzed,andthefixedpointsthatdeterminethecriticalbehaviorofthesystemarefound. Itisshown
a that the effect of elastic deformations on a system with a long-range interaction causes changes in
J
its critical, as well as multicritical, behavior.

6
2 PACSnumbers: 64.60.-i

]
h
In compressible systems, the relation of the order pa- into accountthe effect of the long-rangeinteractionwith
c
rameter to elastic deformations plays an important role. different values of the parameter σ.
e
m Earlier [1] it was shown that, in the case of an elasti- ForahomogeneousIsing-likemodelwithelasticdefor-
cally isotropic body, the critical behavior of a compress- mations and a long-range interaction, the Hamiltonian
-
t ible system with a quadratic striction is unstable with can be represented in the form
a
respect to the relation of the order parameter to acous-
t
s tic modes, and a first-order phase transition close to a
. 1 u
mat sfoercmonudla-oterddeirnotnheeicsitreedalpizaepde.r [H1]owhoelvdero,ntlhyefocronlocwlupsiroenss- H0 =Z dDx[2(τ1+∇σ)S~(x)2++40!(S~(x)2)2]
sures. As was shown later [2], at high pressures begin- 3 3
d- ning from a certaintricritical point Pt, the deformations +Z dDx[a1( uαα(x))2+a2 u2αβ]+
n induced by the external pressure affect the system to a αX=1 αX,β=1
o greaterextentandleadtoachangeofsignoftheeffective 3
1
c interactionconstantforthe orderparameterfluctuations + a dDxS~(x)2( u (x)) (1)
[ 2 3Z αα
and, as a consequence, to a change in the order of the αX=1

1 phasetransition. Inthiscase,accordingto[2],ahomoge-
v neous compressible system is characterizedby two types whereS(x)isthescalarorderparameter,u0 isapositive
6 of tricritical behavior with a fourth-order critical point constant,τ0 ∼|T−Tc|,Tc isthephasetransitiontemper-
2 formed as the point of intersection of the two tricritical ature, uαβ is the strain tensor, a1 and a2 are the elastic
6
1 curves. Calculations performed in terms of the two-loop constantsofthe crystal,anda3 is the quadraticstriction
0 approximation[3]confirmedthe presenceoftwotypesof parameter. Let us change to the Fourier transforms of
5 tricriticalbehaviorforIsing systemsandprovidedvalues the variablesin Eq. (1) andperformintegrationwith re-
0 of the tricritical indices. specttothecomponentsdependingonthenonfluctuating
t/ In structural phase transitions that occur in the ab- variables, which do not interact with the order param-
a eter S(x) Then, introducing, for convenience, the new
senceofthepiezoelectriceffect,intheparaphasetheelas-
m 3
tic strains play the role of a secondary order parameter
variable y(x) = u (x), we obtain the Hamiltonian
- whose fluctuations are not critical in most cases. αα
d αP=1
n The effect of the long-range interaction described by of the system in the form
o the power law 1/r−D−σ at long distances was studied
1
c analytically in terms of the ε-expansion [46] and numer- H = dDq(τ +qσ)S S
: 0 2Z 0 q −q
v ically by the Monte Carlo method [79] in two and three
Xi dcoimnseindseiroanbsl.yIatffwecatssfothuendcrtithiacatltbheehlaovnigo-rraonfgIesiningtesryasctteimons +u40! Z dDqiSq1Sq2Sq3S−q1−q2−q3+
ar for the parameter values σ < 2. A recent study carried a(0)
outforathree-dimensionalspaceinthetwo-loopapprox- +a dDqy S S + 3 y dDqS S
3Z q1 q2 −q1−q2 Ω 0Z q −q
imation [10] confirmed the prediction of the ε-expansion
for systems with long-range interactions. 1 1a(0)
+ a dDqy y + 1 y2 (2)
This paper describes the critical and tricritical behav- 2 1Z q −q 2 Ω 0
ior of three-dimensional compressible systems by taking
In Eq. (2), the components y describing the homoge-
0
neousstrainsareseparated. Accordingto[1],suchasep-
aration is necessary, because the inhomogeneous strains
∗Electronicaddress: [email protected] y are responsible for the acoustic phonon exchange and
q
2
lead to long-range interactions, which are absent in the for the vertex parts of the irreducible Green functions.
caseofhomogeneousstrains. Forthesystemunderstudy, To calculate the β− and γ-functions as the functions
let us determine the effective Hamiltonian that depends involved in the CallanSymanzik equation for renormal-
only on the strongly fluctuating order parameter S ized interaction vertices u, a , and a(0), or complex
1 1
vertices z = a2/4a , w = a(0)2/4a(0), and v = u12z,
1 3 1 3
exp{−H[S]}=BZ exp{−HR[S,y]} dyq (3) which are more convenient for the determination of crit-
Y
ical and tricritical behavior, a standard method based
If the experiment is performed at constant volume, y is on the Feynman diagram technique and on the renor-
0
a constant and the integration in Eq. (3) is performed malization procedure [13] was used with the propagator
with respect to only the inhomogeneous strains, while G(~k)=1/τ +|~k|σ. As aresult,thefollowingexpressions
the homogeneous strains do not contribute to the effec- wereobtainedfortheβ−andγ-functionsinthetwo-loop
tive Hamiltonian. If the experiment occurs at constant approximation:
pressure, the term PΩ is added to the Hamiltonian, the
volume is represented in terms of the strain tensor com-
ponents as
Ω=Ω [1+ u + u u +O(u3)] (4)
0 αα αα ββ
αX=1 αX6=β β = −(2σ−D)v 1−36vJ +1728 2J −J2− 2G v2 ,
v h 0 (cid:16) 1 0 9 (cid:17) i
and the integration in Eq. (3) is performed with respect
β = −(2σ−D)z 1−24vJ −2zJ
tothehomogeneousstrainsaswell. Accordingto[11],the z h 0 0
inclusionofquadratictermsinEq. (4)maybeimportant 2
+ 576(2J −J2− G)v2 ,
whendealing withhighpressuresandcrystalswith large 1 0 3 i
strictioneffects. Theneglectofthesequadratictermsre-
β = −(2σ−D)w 1−24vJ −4zJ +2wJ
strictsthe applicability ofthe results obtainedbyLarkin w h 0 0 0
and Pikin [1] to the case of low pressures. Thus, the 2
+ 576(2J −J2− G)v2 .
Hamiltonian has the form 1 0 3 i
1 γ = (2σ−D) −12vJ −2zJ +2wJ
H = 2Z dDq(τ0+qσ)SqS−q t h 0 0 0
1
+(u − z0) dDq dDq dDq S S Sa S + 288(cid:16)2J1−J02− 3G(cid:17)v2i,
0 2 Z 1 2 3 q1 q2 q3 −q1−q2−q3 γ = (2σ−D)192Gv2,
ϕ
1
+ (z −w ) dDq dDq S S S S , (5) dDqdDp
2Ω 0 0 Z 1 2 q1 −q1 q2 −q2 J = ,
1 Z (1+|~q|σ)2(1+|p~|σ)(1+|q2+p2+2p~~q|σ/2)
z =a2/(4a ), w =a(0)2/(4a(0)),
0 1 3 0 1 3 dDq
J = ,
The effective interaction parameter v = u −z /2 that 0 Z (1+|~q|σ)2
0 0 0
appearsintheHamiltonianduetostrictioneffects,which ∂
G = − dDqdDp(1+|q2+k2+2~k~q|σ)−1
are determined by the parameter z0, can take not only ∂|~k|σ Z
positivebutalsonegativevalues. Asaresult,the Hamil-
tonian describes both first-order and secondorder phase (1+|p~|σ)−1(1+|q2+p2+2p~~q|σ/2)−1
transitions. At v = 0, the system exhibits a tricriti-
0
cal behavior. In its turn, the effective interaction de-
termined inHamiltonian(5) by the parameterdifference
z w may cause a second-order phase transition in the
0 0
system when z w >0 and a first-order phase transition
0 0 Redefining the effective interaction vertices as
when z w < 0. This form of the effective Hamiltonian
0 0
suggeststhatahigherordercriticalpointcanberealized
inthesystemasthepointofintersectionofthetricritical
curves when the conditions v = 0 and z = w are si-
0 0 0
multaneously satisfied [2]. It should be noted that, with
the tricritical condition z0 = w0, Hamiltonian (5) of the v1 =v·J0, v2 =z·J0, v3 =w·J0. (6)
modelunderconsiderationisisomorphicwiththeHamil-
tonian of a rigid homogeneous system.
Intheframeworkofthefield-theoreticalapproach[12],
the asymptotic criticalbehaviorand the structure of the
phase diagrams in the fluctuation region are determined
by the CallanSymanzik renormalization group equation we arrive at the following expressions for the β− and γ-
3
functions: Inthetwo-loopapproximation,the functionswerecal-
culated using the [2/1] approximant. The character of
β = −(2σ−D)v 1−36v (7)
1 1 1 the critical behavior is determined by the existence of a
h
2 stable fixed point satisfying the set of equations
+1728 2J −1− G v2 ,
(cid:16) 1 9 (cid:17) 1i
β (v∗,v∗,v∗)=0 (i=1,2,3). (9)
f e i 1 2 3
β = −(2σ−D)v 1−24v −2v
2 2 1 2
h
Therequirementthatthefixedpointbestableisreduced
2
+576(2J1−1− 3G)v12i, to the condition that the eigenvalues bi of the matrix
β3 = −(2σ−fD)v3h1−2e4v1−4v2+2v3 Bi,j = ∂βi(v1∗,v2∗,v3∗). (10)
∂v
2 j
+576(2J −1− G)v2 ,
1 3 1i lieinthehalf-planeoftheright-handcomplex. Thefixed
γ = (2σ−Df) −12v −e2v +2v point with v∗ = 0, which corresponds to the tricritical
t 1 2 3
h behavior, is a saddle point and must be stable in the
1
+288 2J −1− G v2 , directions determined by the variables z and w and un-
(cid:16) 1 3 (cid:17) 1i stableinthedirectiondeterminedbythevariablev. The
γϕ = (2σ−Df)192Gv12. e stabilization of the tricritical fixed point in the direction
determined by the variable v is achieved by taking into
The redefining procedureemakes sense for a σ ≤ D/2.
account the sixth-order terms with respect to the order
In this case, J , J , and G become divergent functions.
0 1 parameterfluctuationsintheeffectiveHamiltonianofthe
Introducing the cutoff parameter Λ and considering the
model. Thefixedpointwithz∗ =w∗,whichcorresponds
ratios J /J2, G/J2 in the limit of Λ → ∞, we obtain
1 0 0 tothetricriticalbehaviorofthesecondtype,isalsoasad-
finite expressions.
dlepointandmustbestableinthedirectionsdetermined
The values of the integrals were determined numeri-
bythevariablesv andz andunstableinthedirectionde-
cally. For the case a a ≤ D/2, a sequence of the values
termined by the variable w. Its stabilization is possible
of J /J2 and G/J2 was constructed for different Λ and
1 0 0 at the expense of the anharmonic effects.
then approximated to infinity.
Theresultingsetoftheresummedβ functionscontains
It is well known that perturbative series expansions
a wide variety of fixed points. The table specifies the
are asymptotic and the interaction vertices of the order
fixed points that are of most interest for describing the
parameter fluctuations in the fluctuation region are suf-
critical and tricritical behavior and lie in the physical
ficiently large to directly apply Eqs. (7). Therefore, to
region of vertex values with v, z, w ≥ 0. The table
extract the necessary physical information from the ex-
also shows the eigenvalues of the stability matrix for the
pressions derived above, the PadeBorel method general-
corresponding fixed points and the critical indices ν and
ized to the three-parameter case was used. The corre-
η.
sponding direct and inverse Borel transformations have
The analysis of the values and stability of the critical
the form
points suggests the following conclusions. Qualitatively,
f(v1,v2,v3)= ci1,i2,i3v1i1v2i2v3i3 thecriticalphenomenaseemtobeidenticalforanyvalue
i1,Pi2,i3 of the long-range interaction parameter a. The critical
∞
= e−tF(v t,v t,v t,v t)dt, behavior of incompressible systems is unstable with re-
1 2 3 4
R0 c spect to the deformation degrees of freedom (points 1).
F(v1,v2,v3)= (i +i1i,i2+,i3i )!v1i1v2i2v3i3. Thestablepointprovestobetheoneataconstantstrain
i1,Pi2,i3 1 2 3 (points 2). Fixed points 3 describe the first type of tri-
ForananalyticalcontinuationoftheBoreltransformof critical behavior of compressible systems, which occurs
afunction,aseriesinanauxiliaryvariable isintroduced: at constant pressure. Fixed points 4 are tricritical for
systemsstudiedatconstantvolume. Points5arefourth-
∞ ordercritical points at which two tricriticalcurves inter-
c
F˜(v ,v ,v ,θ)= θk i1,i2,i3vi1vi2vi3δ , sect.
1 2 3 k! 1 2 3 i1+i2+i3,k
Xk=0 i1X,i2,i3 For the tricritical behavior of the first type (points 3),
(8) Hamiltonian (5) is isomorphic with the Hamiltonian of
and the [L/M] Pade approximationis applied to this se- an incompressible homogeneous model and, hence, the
ries at the point θ = 1. This approach was proposed critical indices also coincide with those of the incom-
and tested in [14] in describing the critical behavior of pressible model. The tricritical behavior of the second
systems characterized by several vertices of interaction type (points 4) corresponds to the critical behavior of
of the order parameter fluctuations. The property [14] a spherical model and is determined by the correspond-
ofthe systemretainingits symmetry underthe Padeap- ing indices. The fourth-order fixed points (points 5) are
proximantsinthe variable is essentialinthe description characterized by the field-average values of the critical
of multivertex models. indices.
4
The large values of the effective vertices z and w in effect of elastic deformations on the critical behavior of
comparison with the systems with short-range interac- systems with a long-range interaction. This effect mani-
tions [3] are caused by the fact that the mechanism gov- festsitselfasachangeinthevaluesofthe criticalindices
erning the effect of elastic deformations on the critical forIsingsystems alongwith the appearanceofmulticrit-
phenomena is related to the dependence of the interac- ical points in the phase diagrams of the substances.
tion integral in the Ising model on the distance between The work is supported by Russian Foundation for Ba-
the lattice sites. sic Research N 04-02-16002.
The study described above revealed the considerable
[1] A.I.LarkinandS.A.Pikin,Zh.ksp.Teor.Fiz.56,1664 [9] E. Luijten and H. W. J. Blote, Phys. Rev. B 56, 8945
(1969) [Sov.Phys. JETP 29, 891 (1969)]. (1997).
[2] Y.Imry,Phys. Rev.Lett.33 (21), 1304 (1974). [10] S. V. Belim, Pisma Zh. ksp. Teor. Fiz. 77, 118 (2003)
[3] S. V. Belim and V. V. Prudnikov, Fiz. Tverd. Tela (St. [JETP Lett. 77, 112 (2003)].
Petersburg) 43, 1299 (2001) [Phys. Solid State 43, 1353 [11] D.J.BergmanandB.I.Halperin,Phys.Rev.B13,2145
(2001)]. (1976).
[4] M. E. Fisher, S.-K. Ma, and B. G. Nickel, Phys. Rev. [12] D. Amit, Field Theory the Renormalization Group and
Lett.29, 917 (1972). Critical Phenomena (McGraw-Hill, New York,1976).
[5] J. Honkonen,J. Phys. A 23, 825 (1990). [13] J.Zinn-Justin,QuantumFieldTheoryandCriticalPhe-
[6] E.Luijten and H.Mebingfeld, Phys.Rev.Lett.86, 5305 nomena (Clarendon Press, Oxford,1989).
(2001). [14] A.I.SokolovandK.B.Varnashev,Phys.Rev.B59,8363
[7] E. Bayong and H. T. Diep, Phys. Rev. B 59, 11 919 (1999).
(1999).
[8] E. Luijten, Phys.Rev. E60, 7558 (1999).
5
N v1∗ v2∗ v3∗ b1 b2 b3 ν η
σ=1.9
1 0.042067 0 0 0.684 -0.184 -0.184 0.652688 0.113420
2 0.044353 0.095190 0 0.684 0.185 0.183 0.751433 0.113420
3 0.044353 0.095190 0.095190 0.684 0.185 -0.185 0.652688 0.113420
4 0 0.5 0 -1 1 1 1.052632 0.1
5 0 0.5 0.5 -1 1 -1 0.526316 0.1
σ=1.8
1 0.023230 0 0 0.628 -0.488 -0.488 0.636349 0.207461
2 0.023230 0.245404 0 0.628 0.489 0.490 0.782912 0.207461
3 0.023230 0.245404 0.245404 0.628 0.489 -0.489 0.636349 0.207461
4 0 0.5 0 -1 1 1 1.111111 0.2
5 0 0.5 0.5 -1 1 -1 0.555556 0.2
σ=1.7
1 0.020485 0 0 0.699 -0.532 -0.532 0.667452 0.304862
2 0.020485 0.266497 0 0.699 0.533 0.532 0.830648 0.304862
3 0.020485 0.266497 0.266497 0.699 0.533 -0.533 0.667452 0.304862
4 0 0.5 0 -1 1 1 1.176471 0.3
5 0 0.5 0.5 -1 1 -1 0.588235 0.3
σ=1.6
1 0.015974 0 0 0.874 -0.616 -0.616 0.697361 0.403936
2 0.015974 0.309684 0 0.874 0.617 0.620 0.889473 0.403936
3 0.015974 0.309684 0.309684 0.874 0.617 -0.618 0.697361 0.403936
4 0 0.5 0 -1 1 1 1.25 0.4
5 0 0.5 0.5 -1 1 -1 0.625 0.4