Crossover from Goldstone to critical fluctuations: Casimir forces in confined O${\bf(n)}$ symmetric systems

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by  Volker Dohm

Crossover from Goldstone to critical fluctuations: Casimir forces in confined O(n)
symmetric systems
Volker Dohm
Institute for Theoretical Physics, RWTH Aachen University, D-52056 Aachen, Germany
(Dated: 11 January 2013)
We study the crossover between thermodynamic Casimir forces arising from long-range fluctua-
3 tionsduetoGoldstonemodesandthosearisingfromcriticalfluctuations. Bothtypesofforcesexist
1 in the low-temperature phase of O(n) symmetric systems for n > 1 in a d-dimensional Ld−1×L
k
0 slab geometry with afiniteaspect ratio ρ=L/L . Ourfinite-size renormalization-group treatment
k
2
for periodic boundary conditions describes the entire crossover from the Goldstone regime with a

n nonvanishingconstant tailof thefinite-sizescaling function farbelow Tc uptotheregion farabove
a Tc including the critical regime with a minimum of the scaling function slightly below Tc. Our
J analytic result for ρ ≪ 1 agrees well with Monte Carlo data for the three-dimensional XY model.

2 A quantitativeprediction is given for thecrossover of systems in theHeisenberg universality class.
1
PACSnumbers: 05.70.Jk,64.60.-i,75.40.-s

]
h
c In the past two decades, substantial effort has been de- film geometry studied earlier [6, 9, 10, 12–14]) is the ab-
e
voted to the study of thermodynamic Casimir forces [1] sence of singularities of the free energy density at finite
m
that result from two fundamentally different sources in temperatures and the existence of a discrete mode spec-
-
confined condensed matter systems: (i) from classical trumwithadominantlowestmodethatisamenabletoa
t
a fluctuations with long-ranged correlations due to mass- simultaneous analytic treatment of the low-temperature
t
s less Goldstone modes [2, 3], and (ii) from long-ranged and the critical regions.
.
t critical fluctuations [4]. Both types of fluctuations exist
a inthelow-temperaturephaseofO(n)symmetricsystems Our analytic treatment is based on the O(n) symmetric
m isotropic ϕ4 Hamiltonian
such as superfluids [5], superconductors [6], XY mag-
- nets (n = 2) [3, 7, 8], and isotropic Heisenberg magnets
d r 1
n (n=3) [7]. While successful analytic theories have been H = ddx 0ϕ2+ (∇ϕ)2+u0(ϕ2)2 (1)
2 2
o developedthatseparatelydescribesuchCasimirforcesei- Z
c ther(i)intheGoldstone-dominatedregimedeeplyinthe V (cid:2) (cid:3)
[ low-temperaturephase[2,3]or(ii)abovebulkcriticality where ϕ(x) = V−1Σkϕkeikx is an n-component field in
1 [9–12],thereis aseriouslackofknowledgeconcerningan a finite volume V =Ld−1L with periodic boundary con-
v analytic theory of the crossover between these two types ditions. The summatiokn Σk runs over discrete k vectors
6 ofCasimirforcesinthelow-temperaturephase. Thegoal including k = 0 up to some cut off Λ. The fundamen-
6 ofthisLetteristoprovidesuchacrossovertheoryforthe
tal quantity from which the Casimir force per unit area
6
casewheretheconfininggeometryhasperiodicboundary
2 conditions. Such systems are well accessible to numeri- FCas = −∂[Lfex]/∂L can be derived is the excess free
. energy density (divided by k T) fex =f −f where
1 cal studies [3, 7, 8]. We shall present analytic results B b
0 for general n that are in very good agreement with the
3 existing Monte Carlo (MC) data for n = 2 both in the f(T,L,Lk)=−V−1ln Dϕexp(−H) (2)
1
Goldstone regime as well as in the critical regioninclud- Z
:
v ing the crossoverbetween these regions. We also present and f ≡ lim f are the free energy densities of the
Xi aquantitativepredictionforthiscrossoveroftheCasimir finitebsystemVan→d∞the bulk system, respectively.
force in the (n = 3) Heisenberg universality class. The
r
a concept of our theory should be applicable also to the It is expected that, for isotropic systems near criticality
case of Dirichlet boundary conditions which are relevant and for large L and Lk, FCas can be written in a finite-
to the crossover of Casimir forces in superfluids [5] and size scaling form [15]
superconductors [6].
F (t,L,L )=L−dX(x˜,ρ) (3)
Cas k
A crucial ingredient of our approach for general n is an
appropriate choice of the geometry. We consider a finite with the scaling variable x˜ = t(L/ξ )1/ν, t = (T −
0+
ddimensionalLd−1×Lslabgeometrywithafiniteaspect T )/T where ξ is the amplitude of the bulk correla-
k c c 0+
ratio ρ=L/L . This is well justified by the fact that all tion length above T . As noted for the case of film ge-
k c
experiments and computer simulations were performed ometry (ρ → 0) [3], the scaling form (3) applies to the
in slab geometries with a small but finite ρ rather than low-temperature region where the scaling function satu-
ρ = 0. Recent Monte Carlo data for ρ = 1/6 [3] and ratesatanonzeronegativevalueX(−∞,0)<0forn>1
ρ=0.01[8] show thatthe ρ - dependence ofthe Casimir [2, 3, 14]. So far, however, no analytic calculation of the
force is quite weak for ρ ≪ 1. The basic advantage of a function X(x˜,0) for systems with n > 1 and periodic
finite-slabgeometry(ratherthanthe idealized∞d−1×L boundary conditions has been performed that describes
X inthewholelow-temperatureregion−∞≤x˜≤0[16].
2
It is the goal of this paper to describe the full crossover The second approximation is to neglect the effect of the
in terms of a single scaling function X(x˜,ρ), for general higher-mode parts ∝ Φσ σ2 and ∝ (σ2)2 on the finite-
L
n > 1 and small ρ, from the Goldstone-dominated be- size properties (but not on the bulk critical exponents
haviorforx˜→−∞uptothe high-temperaturebehavior which will be incorporated via Borel-resummed field-
for x˜ ≫ 1 including the critical region 0 ≤ |x˜| ∼ O(1) theoretic functions). No further approximation (such
above and below T . as an ε = 4 − d expansion) will be made in the sub-
c
sequentrenormalization-grouptreatmentatfixedd. Our
Animportantconceptualdifferencebetweentheprevious approach goes beyond a naive Gaussian approximation
perturbation approach within the ϕ4 theory for (n = 1) not only because of the fourth-order term ∼ u (Φ2)2
0
systemswithadiscrete(Ising-like)symmetry[11]andfor
in H but also because of the size-dependent couplings
0
systems with a continuous symmetry (n > 1) presented ∼ u M2σ 2 and ∼ u M 2σ 2 between the lowest-mode
in this paper is the following. For n = 1, two sepa- 0 0 L 0 0 T
averageandthehighermodeswhicharisefromtheterms
rate perturbationapproacheswerenecessaryfor describ-
r¯ σ 2 and r¯ σ 2 , respectively. After integration over
0L L 0T T
ingfinite-sizeeffectsbothinthecentralfinite-sizeregime
σ, we obtain the unrenormalized free energy density
near criticality and those in the low-temperature phase
farbelowT inordertocapturethetwo-fold(spinupand
c
spindown)degeneracycharacteristicofthe groundstate
ofIsing-likesystems. Forn>1,thereexistbothlongitu-
dinal and transverse fluctuations. The latter correspond
1
toorientationalchangesthatpermittheorderparameter f =f − ln dnΦexp[−H (Φ)]
0 0
V
fluctuations to exhaust the full phase space since there
nZ o
exists no large barrier like that between the purely lon- 1 n−1
+ S (r¯ )+ S (r¯ ), (6)
0 0L 0 0T
gitudinal spin-up and spin down configurations of large 2 2
Ising-like systems far below T . As a consequence, a sin-
c
gleperturbationansatzsufficesforn>1tocaptureboth
the critical fluctuations and the fluctuations due to the
Goldstone modes. This results in a smooth description
of the crossover between the two different regimes with- with S (r) = V−1 ln(r+k2) where f is indepen-
0 k=6 0 0
out the necessity of matching two separate pieces of the dent of r and u .
0 0
P
theory.
For finite V, the transverse parameter r¯ remains posi-
0T
We decompose ϕ = Φ+σ into a homogeneous lowest- tive in the whole range −∞ < r < ∞ and interpolates
0
mode Φ = V−1 ddxϕ(x) and higher-mode fluctua- smoothly between the large-volume limit above and be-
σti(oxn)siσnt(ox)”l=ongVit−u1RdΣinka=6 l0”ϕaknedikx”.traWnsevefursret”hepradrtescoσm(xp)os=e low Tc,
σ (x)+σ (x) whichareparallelandperpendicularwith
L T
respect to Φ. Correspondingly, the Hamiltonian H is
decomposed as H =H (Φ)+H(Φ,σ) where
0
r for r ≥0,
H0(Φ)=V 21r0Φ2e+u0(Φ2)2 , (4) Vl→im∞r¯0T =(cid:26) 00 for r00 ≤0. (7)
(cid:20) (cid:21)
1
H(Φ,σ)= ddx r σ 2+r σ 2+(∇σ )2
0L L 0T T L
2
VZ (cid:8) (cid:2) Thevanishingofthetransverseparameterr¯0Tisthechar-
e +(∇σ )2 +4u Φσ σ2+u (σ2)2 (5) acteristicsofthemasslessGoldstonemodeswhicharethe
T 0 L 0
origin of long-range correlations and the Casimir force
with the longitudinal (cid:3)and transverse para(cid:9)meters well below T . In higher-order perturbation theory for
c
r (Φ2) = r +12u Φ2,r (Φ2)= r +4u Φ2. The free the bulk system, spurious (infrared) singularities arise
0L 0 0 0T 0 0
energy density f is then calculated by first integrating due to the vanishing of r¯ [17]. Within our approxima-
0T
over σ and subsequently over Φ. tion, such spurious singularities do not yet appear since
S (r¯ ) has a finite large-volume limit below T .
0 0T c
An important reference quantity of our theory
is the n-dependent lowest-mode average M2 = The bare expression (6) does, of course, not yet cor-
0
dnΦΦ2exp[−H (Φ)]/ dnΦexp[−H (Φ)]. The phys- rectly describe the crossover from the Goldstone to the
0 0
ical importance of the quantity M2 is related to the critical regime. Both additive and multiplicative renor-
R R 0
fact that the main contribution of the integration over malizationsarenecessary,aftersubtractinganonsingular
Φ comes from the region around Φ2 ≈M2 and that this bulk part f . Integration of the renormalization-group
0 ns
quantity is relevant in the whole range −∞ < r < ∞ equation then leads to the scaling form f (t,L,L ) =
0 s k
above and below Tc. This provides the justification L−dF(x˜,ρ) for the singular part fs = f −fns. We have
for replacing the Φ2-dependent parameters r0L(Φ2) and performed these steps within the minimal renormaliza-
r (Φ2) in H by r¯ ≡ r (M2) and r¯ ≡ r (M2). tion scheme at fixed dimensions 2 < d < 4 [18]. The
0T 0L 0L 0 0T 0T 0
e
3
result reads 0
-0.1 n = 2
˜ld ν Q∗2x˜2˜l−α/ν
F(x˜,ρ)=−A + B(u∗) -0.2
d
4d 2α
( -0.3
(n−1) l2 ld/2 -0.4
− ε "4T˜lε − Td #) X-0.5
-0.6
n 2π2A1/2
+ρd−1(− 2 ln ˜lε/2ρ(d−1)/2[Γ(nd/2)]2/nu∗1/2 --00..87 ( a ) ρρ == 11//66 dM=C3 XthYeo mryo del
(cid:16) (cid:17) ρ = 0 ε expans. GD
∞ -0.9 ρ = 0 ε expans. KD
− ln 2 dssn−1 exp − 1y˜(x˜,ρ)s2−s4 ρ = 0 Goldstone
-1
2 -15 -10 -5 0 5 10 15
(cid:16) Z0 (cid:2) (cid:3)(cid:17) t (L / ξ )1/ν
0+
1 n−1
+ J0(˜l2,ρ)+ J0(lT,ρ) , (8) 0 ρ = 1/6 d = 3 theory
2 2 ) ρ = 1/6 MC Heisenberg model
-0.2 ρ = 0 ε expans. GD
ρ = 0 ε expans. KD
ρ = 0 Goldstone
-0.4
∞
J (z,ρ)=ρ1−d dyy−1 exp −zy/(4π2) -0.6
0
( X
Z
0 (cid:2) (cid:3) -0.8
d−1
× (π/y)d/2− ρK(ρ2y) K(y)+ρd−1 -1
(cid:20) h i (cid:21) -1.2 ( b ) n = 3
−ρd−1exp(−y) (9)
-1.4
)
-15 -10 -5 0 5 10 15
with K(y)= ∞ exp(−ym2) and with t (L / ξ )1/ν
m=−∞ 0+
l (x˜,ρ)=P˜l2−8˜lε/2u∗1/2ρ(d−1)/2A−1/2ϑ (y˜), (10) FIG.1: (Coloronline)ScalingfunctionX(x˜,ρ)oftheCasimir
T d 2
force as a function of x˜ = t(L/ξ0+)1/ν (a) for n = 2 and (b)
for n = 3 in three dimensions. Solid lines: d = 3 theory for
∞ ∞
ρ = 1/6 as calculated from (8),(13). Dot-dashed and dotted
ϑ2(y)= dssn+1e−21ys2−s4/ dssn−1e−21ys2−s4, lines: εexpansionresultsforρ=0[9,10]. Horizontaldashed
Z Z lines: Casimir amplitude X(−∞,0) = −(n−1)ζ(3)/π. MC
0 0
(11) data (a) from [3] for the XY model and (b) from [7] for the
where ˜l(x˜,ρ) and y˜(x˜,ρ) are determined implicitly by Heisenberg model.
y˜+12ϑ (y˜)=ρ(1−d)/2˜ld/2A1/2u∗−1/2, (12a)
2 d To obtainthe scalingfunction X ofthe Casimirforce we
y˜= x˜Q∗ ˜l−α/(2ν)ρ(1−d)/2A1/2u∗−1/2. (12b) considerthe singularpartofthe bulkfree energydensity
d f±(t)= A±|t|dν. It can be written as f± = L−dF±(x˜)
s,b s,b b
The fixed point value of the renormalized four-point where F±(x˜) is the bulk part of F(x˜,ρ). It is derived
coupling u is denoted by u∗. The quantities Q∗ = from (8)bin the limit of large |x˜|. The scaling function of
Q(1,u∗,d) and B(u∗) are the fixed point values of the fex is then given by Fex(x,ρ) = F(x,ρ)−F±(x) from
b
ndependentamplitude functionQ(1,u,d)ofthesecond-
which we calculate the desired scaling function
momentbulkcorrelationlengthaboveT andofthefield-
c
theoreticfunctionB(u)relatedtotheadditiverenormal- x˜∂Fex(x˜,ρ) ∂Fex(x˜,ρ)
X(x˜,ρ)=(d−1)Fex(x˜,ρ)− −ρ (.13)
ization of bulk theory [18], respectively. A = Γ(3 −
d ν ∂x˜ ∂ρ
d/2)[2d−2πd/2(d−2)]−1 is an appropriate geometric fac-
tor. In the limit ρ→0,x˜→−∞, our function X(x˜,ρ) yields
the finite low-temperature amplitude
Eq. (8) is valid for general n and 2 < d < 4 above, at,
and below Tc including the Goldstone regime for n > 1. X(−∞,0)=−(n−1)(d−1)π−d/2Γ(d/2)ζ(d) (14)
It incorporatesthe correctbulk critical exponents α and
ν andthecompletebulkfunctionB(u∗)(notonlyinone- oftheGoldstoneregimeinfilmgeometry. Forn=2,d=
looporder). F(x˜,ρ)isananalyticfunctionofx˜atfiniteρ,
3 this agrees with Eq. (25) of [3].
in agreement with general analyticity requirements. For
n=1,thefunction(8)isnotapplicabletotheregionwell MC data are available for both the three-dimensional
below T as it does not capture the two-fold degeneracy XY [3] and Heisenberg [7] models with the aspect ratio
c
of the ground state of Ising-like systems, as discussed in ρ = 1/6. The comparison of our result for ρ = 1/6 with
[11]. the MC data is shown in Figs. 1 (a) and (b) for n = 2
4
and n =3, respectively. Here we have employed the fol- ofapplicabilityofourtheoryatfiniteρ. Forthispurpose
lowing numerical values [19, 20]: ν = 0.671,0.705, α = weconsiderthefinite-sizeamplitudeF(0,ρ)ofEq. (8)at
−0.013,−0.115, u∗ = 0.0362,0.0327, Q∗ = 0.939,0.937, T =T . Weexpectamonotonicρ-dependenceofF(0,ρ)
c
B(u∗) = 1.005,1.508, for n = 2,3, respectively. Also onthebasisofthemonotonicityhypothesisinferredfrom
shown are the ε expansion results for ρ = 0 from [9, 10] previous results for n = 1 [11, 21] and n = ∞ [11]. A
for T ≥ T (dot-dashed and dotted lines). The horizon- comparisonwith MC data for the three-dimensional XY
c
tal dashed lines represent the ρ = 0 Casimir amplitude model [22] is shown in Fig. 2 in the range 0 ≤ ρ ≤ 1/2.
(14) due to the Goldstone modes. For n = 2 there is TheMCdatashowaveryweakρ-dependenceforρ≪1.
an excellent overall agreement of our result (solid line) Ourresultis inreasonableagreementwiththe MC data,
with the MC data [3] including the crossover from the exceptthatthemonotonicityoftheρ-dependenceisnot
Goldstone-dominated region to the critical region. reproduced by our theory for small ρ < ρ = 0.196
max
(dotted portion of the curve). The deterioration of the
For n = 3 [Fig. 1 (b)] there are systematic deviations quality of our theory for ρ → 0 is to be expected since
betweenall theoreticalcurvesand the MC data of[7]. A the separation between the lowest mode and the higher
similar discrepancy with the MC data of [7] exists also
modes goes to zero in this limit. For this reason it is
forn=2[notshowninourFig. 1(a)]asshowninFig. 6
expected that Eqs. (8) - (12) are not quantitatively re-
of[3]fortheXYmodel. Asnotedin[3],thesediscrepan-
liable for ρ ≪ 0.2 in some region |x˜| ≪ 1 close to T .
c
ciesmightbeduetotheuncertaintyinthenormalization
In particular,the scaling function of the film free energy
factor used for the MC data in [7]. More accurate MC
density F (x˜) = lim F(x˜,ρ) obtained from (8) is
film ρ→0
simulations are desirable in order to test our prediction
not expected to be reliable for |x˜| ≪ 1. This function
for n=3. has, in fact, an artificial cusp-like singularity at x˜ = 0
similar to that of previous approximate ρ = 0 theories
0
[6, 9, 10, 12–14]. Our monotonicity criterion provides
n = 2
d = 3 theory
a theory-internal argument against the reliability of ap-
-0.1 extrapolation
proximate ρ=0 results in the region |x˜|≪1. Neverthe-
MC XY model
ρ) less, assuming a negligible ρ - dependence for ρ < ρmax
0, -0.2 weobtainfromtheextrapolationofthemaximuminFig.
F( 2 to ρ=0 (dashed line in Fig. 2) our prediction
-0.3
F (0)≈F(0,ρ )=−0.3275 (15)
-0.4 film max
-0.5 forfilmgeometryatbulkTc. Thisisinacceptableagree-
ment with the MC estimate −0.292 at ρ = 0.01 [22]
0 0.1 0.2ρ 0.3 0.4 0.5 shown in Fig. 2.
It would be interesting and important to extend our ap-
FIG. 2: (Color online) Critical amplitude F(0,ρ), (8), for proach to the most relevant case of Dirichlet boundary
n=2and d=3at Tc (solid line). Themaximum −0.3275 is conditions in order to explain the experimental results
at ρmax = 0.196. The dashed line is the extrapolation from fortheCasimirforceinsuperfluid4He[5]forwhichthere
ρ=ρmaxtoρ=0(filmgeometry). Thedottedlinerepresents exists no satisfactory analytic theory so far.
(8)atTc forρ<ρmax. MCestimate(triangles) forthed=3
XY model for ρ=0.01,1/4,1/2 from [22] .
The author is grateful to D. Dantchev, M. Hasenbusch,
and O. Vasilyev for providing the MC data of [3, 7, 22]
Finally we discuss the question as to the expected range in numerical form.
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