- Email: [email protected]
Critical shear viscosity of 4HE above and below tλ
Physica B 165&166 (1990) 563-564 North-Holland CRITICAL
SHEAR VISCOSITY
OF 4HE ABOVE
AND BELOW T
A
R. SCHLOMS, J. PANKERT, V. DOHM fiir
Institut
Theoretische
Physik,
On the basis of a new stochastp and below the h transition of the shear viscosity. Both are malization-group theory. Good
Aachen,
Technische' Hochschule
dynamic model the critical behavior of the shear viscosity above He 1s calculated. We show that two different mechanisms contribute to described by couplings which are irrelevant in the sense of the renoragreement with experiments is found.
1.
INTRODUCTION Detailed measurements have shown /l-4/ that the shear viscosity fl has a weak&y singular behavior near the h transition of He. No quantitheory is available so far for this funtative damental transport coefficient. In the following we present the results of a quantitative renormalization-group (RG) calculation for n both Q via the correlaabove and below Th. We define tion tunction of the components j, of the transj of the momentum density in the verse part limit of small k and w /5/,
< ja j,>
= t
k TaO =
b
24 2-1 ~p2kBTrlk2h2+qk/pn) ,
c@
-
H = HF($,m) where HF(!,m) is the model F Hamiltonian /7/. The Gaussian Langevin forces 8.(x,t) have correlations in accord with the equ!librium distributionwexp - H. If the variable j is dropped we recover model F whose nonuniversal parameters are well known /7,8/. Since x. and g. are also r\ known /6/ we are left with twd new parameters can be determined from the measure a and q1 which Then the calculation below Th does rl above ‘$. not contain any adjustable parameter. The Hamiltonian (7) could be extended to include also critical contributions to pn and p which are neglected here.
(1)
kcc kbik2
3.
CALCULATION AND RESULTS Within our model we can define the vertex tensor ryl(k,w) where response field /9/ Gdnjugate to
(2)
is the normal fluid density (pn=p above where p Th). Th?s correlation function will be calculated from the stochastic equations of a new model that includes reversible and dissipative couplings of j(x,t) to the order parameter $(x,t). Althoug both types of couplings are irrelevant in the RG sense they are nonnegligible and lead to an observable singularity of rl near T in A agreement with experiments.
To leading kBT we find
q
0
in the dissipative part of the two uhenomenological constants we propose the following model j! = -
2r6H/6$”
+ ig$SHIGm
2gIm
+
-
j Q,
equation, and ql.
g.vJ.‘.EH/6j J
(IQ” 6H/6$“)
+ 0
+ Om
4g
j
=
r[qo26H/6j
+ 2 g
j
Re(v$
6H’iQ)
+ fi 1’2
=[aD+
j, kBT].
couplings
ql
and
(8) gj
=
i
[ 2(d-1) kBTl
cx,t)abJi
(x,tf
+ c.c.]/~ ,(Io)
where <...> denotes the Fourier transformed correlation function at k = W = 0, calculated within model F. The parameter r? can be absorbed which we take in the finite value qh of q at $ We have determined the critical from experiment. exponent of the last term of (9) and have calculated its amplitude in one-loop order (for T
q(4)
,(5)
ejl
-
ll)) = rll(
-
, (6)
0921-4526/90/$03.50
of
the
F
rl
.
the
$$k$/k2)<‘rabKy6$
ncrg(x,t)
with Thus
+
in
terms
with
(3)
+
order
in
(9)
MODEL wz have In constructing an appropriate model He as started from the equations of motion for given in Ref. 6. We keep j(x,t), ji(x,t) and the entropy variable m(x,t) and neglect the coupling to the first-sound mode. As a novel feature we include a Q-dependent bare kinetic coefficient
=
q
Jdenotes
/[Z(d-1) q = ak2 Tr[zk Ij3(k,C)]k=0
2.
ij(+)/kBT
Germany
@ 1990 - Elsevier Science Publishers B.V. (North-Holland)
+ g;
A;
[<(t,)
r(t+)
kBT]-l,
(11)
R. Schloms, J. Pankert,
564
A+ = -
r(
(16,&l
.
A; = -
(12)
,
(32rr)-1
= -2t , t = (T-T~)/T~. were where t = t and t r(t) de;otes the known effective renormalized kinetic coetticient ot the order parameter /7, S/ and c(t) is the correlation length above T h Since the Leading singular term of
- :l$l*>A= co
t/t1-
+
4.
V. Dohm
DISCUSSION We have treated
contributions (i)
(13)
Cl?
is known from the specific heat, the result (11) as an unknown parameter. We contains only ql so as to achieve agreement with have adjus:ed ql -3 above Th. The second the data2/!-4/ at t = 10 ,n (11) turns out co be one order ot term --g. and does magnitud2 smaller than the first term to the term not change sign at T . in contrast 113) and to the measbred values ot q - qh. The comparison between theory and the data is shown dependence above Th in Fig. 1. The temperature and both the magnitude and the temperature well described by our dependence below Th are ‘hcory without adjustments.
-1
the the
eftects critical
ok
two
shear
ditterent
viscosity.
The stress-tensor correlation[last term of (9)] is parallel to the known formula ot fluctuating hydrodynamics ot ordinary fluids /lo/. According to th? last term of (11) this eftect alone would yield the wrong sign wrong magnitude of rl - rlh above Th and the below
(ii)Thr
to
TA.
2 I$1 term in (3) models the Ipading dependence ot the ettec ive noncritical visb cosity ot superfluid He on the supertluid traction. Above Th this shows up as a local effect corresp ncilng to tluctuating clusters e of superfluid He, as eculated earl irr ill/. Below TA the ,$/“term contributes significantly already at the hydrodynamic (mean field) level.
The semiquantitative considerations Ferrell and collaborators /12.13/ do into account the effect (ii). Their based on the etfect (i) difter tram with respect to sign and magnitude. that their results provide a correct of the ettcctivr exponent ot q - oh
by not take results ours both We doubt explanation above Th.
*
c Is, 0
-*
-3 -A
I
I
I
-1 *-
-c=
-
g
-2
-
-3
-THEORY
log
Critical shear Data tram Rets. our expression
I
I
I
ItI
FTGURE 1 viscosity $= (Q - QA)/~A vs l-4. The solid lines represent (11) - (13).
t.
REFERENCES C. Howald, and H. Meyer, J. Low Ill S. Wang, Temp. Phys.. to be published. /2/ L. Bruschi, G. Mazzi, M. Santini. and G. Torzo, J. Low Temp. Phys. 29 (1977) 63. /3/ R. Biskeborn and R.W. Guernsey. Phys. Rev. Lett. 34 (1975) 455. /4/ R.W.H. Webeler and G. Allen, Phys. Rev. A 5 (1972) 1820. /5/ P.C. Hohenborg and P.C. Martin, Ann. Phys. (N.Y.) 34 (1965) 291. f6J J. Pankert and V. Dohm. Phys. Rev. B 40 (1989) 10842. 193. /7/ V. Dohm, 2. Phys. B 61 (1985) /8/ W.Y. Tam and G. Ahlers, Phys. Rev. B 33 (1986) 183: B 37 (1988) 7898. /9/ P.C. Martin, E.D. Siggia, and H.A. Rosr,, Phys. Rev. A 8 (1973) 423. /lO/L.D. Landau and E.M. Litschitz. Sov. Phys. JEPT 5 (1957) 512. /11/K. Mendelssohn. in Encyclopedia of Physics, Vol. XV, cd. S. Fliigge (Springer. Brri in, 1956). /12/R.A. Ferrell, J. Low Temp. Phys. 70 (1988) 435. /13/J.K. Bhattacharjer. R.A. Ferrell, and Z.Y. in Proceedings of 17th International Chen, Conierencc on Low Temperaturf, Physics, cds. LJ. Eckern, A. Schmid. W. Weber. and H. Wiihl (North-Holland, New York, 1984). p. 977.
SHEAR VISCOSITY
OF 4HE ABOVE
AND BELOW T
A
R. SCHLOMS, J. PANKERT, V. DOHM fiir
Institut
Theoretische
Physik,
On the basis of a new stochastp and below the h transition of the shear viscosity. Both are malization-group theory. Good
Aachen,
Technische' Hochschule
dynamic model the critical behavior of the shear viscosity above He 1s calculated. We show that two different mechanisms contribute to described by couplings which are irrelevant in the sense of the renoragreement with experiments is found.
1.
INTRODUCTION Detailed measurements have shown /l-4/ that the shear viscosity fl has a weak&y singular behavior near the h transition of He. No quantitheory is available so far for this funtative damental transport coefficient. In the following we present the results of a quantitative renormalization-group (RG) calculation for n both Q via the correlaabove and below Th. We define tion tunction of the components j, of the transj of the momentum density in the verse part limit of small k and w /5/,
< ja j,>
= t
k TaO =
b
24 2-1 ~p2kBTrlk2h2+qk/pn) ,
c@
-
H = HF($,m) where HF(!,m) is the model F Hamiltonian /7/. The Gaussian Langevin forces 8.(x,t) have correlations in accord with the equ!librium distributionwexp - H. If the variable j is dropped we recover model F whose nonuniversal parameters are well known /7,8/. Since x. and g. are also r\ known /6/ we are left with twd new parameters can be determined from the measure a and q1 which Then the calculation below Th does rl above ‘$. not contain any adjustable parameter. The Hamiltonian (7) could be extended to include also critical contributions to pn and p which are neglected here.
(1)
kcc kbik2
3.
CALCULATION AND RESULTS Within our model we can define the vertex tensor ryl(k,w) where response field /9/ Gdnjugate to
(2)
is the normal fluid density (pn=p above where p Th). Th?s correlation function will be calculated from the stochastic equations of a new model that includes reversible and dissipative couplings of j(x,t) to the order parameter $(x,t). Althoug both types of couplings are irrelevant in the RG sense they are nonnegligible and lead to an observable singularity of rl near T in A agreement with experiments.
To leading kBT we find
q
0
in the dissipative part of the two uhenomenological constants we propose the following model j! = -
2r6H/6$”
+ ig$SHIGm
2gIm
+
-
j Q,
equation, and ql.
g.vJ.‘.EH/6j J
(IQ” 6H/6$“)
+ 0
+ Om
4g
j
=
r[qo26H/6j
+ 2 g
j
Re(v$
6H’iQ)
+ fi 1’2
=[aD+
j, kBT].
couplings
ql
and
(8) gj
=
i
[ 2(d-1) kBTl
cx,t)abJi
(x,tf
+ c.c.]/~ ,(Io)
where <...> denotes the Fourier transformed correlation function at k = W = 0, calculated within model F. The parameter r? can be absorbed which we take in the finite value qh of q at $ We have determined the critical from experiment. exponent of the last term of (9) and have calculated its amplitude in one-loop order (for T
q(4)
,(5)
ejl
-
ll)) = rll(
, (6)
0921-4526/90/$03.50
of
the
F
rl
.
the
$$k$/k2)<‘rabKy6$
ncrg(x,t)
with Thus
+
in
terms
with
(3)
+
order
in
(9)
MODEL wz have In constructing an appropriate model He as started from the equations of motion for given in Ref. 6. We keep j(x,t), ji(x,t) and the entropy variable m(x,t) and neglect the coupling to the first-sound mode. As a novel feature we include a Q-dependent bare kinetic coefficient
=
q
Jdenotes
/[Z(d-1) q = ak2 Tr[zk Ij3(k,C)]k=0
2.
ij(+)/kBT
Germany
@ 1990 - Elsevier Science Publishers B.V. (North-Holland)
+ g;
A;
[<(t,)
r(t+)
kBT]-l,
(11)
R. Schloms, J. Pankert,
564
A+ = -
r(
(16,&l
.
A; = -
(12)
,
(32rr)-1
= -2t , t = (T-T~)/T~. were where t = t and t r(t) de;otes the known effective renormalized kinetic coetticient ot the order parameter /7, S/ and c(t) is the correlation length above T h Since the Leading singular term of
t/t1-
+
4.
V. Dohm
DISCUSSION We have treated
contributions (i)
(13)
Cl?
is known from the specific heat, the result (11) as an unknown parameter. We contains only ql so as to achieve agreement with have adjus:ed ql -3 above Th. The second the data2/!-4/ at t = 10 ,n (11) turns out co be one order ot term --g. and does magnitud2 smaller than the first term to the term not change sign at T . in contrast 113) and to the measbred values ot q - qh. The comparison between theory and the data is shown dependence above Th in Fig. 1. The temperature and both the magnitude and the temperature well described by our dependence below Th are ‘hcory without adjustments.
-1
the the
eftects critical
ok
two
shear
ditterent
viscosity.
The stress-tensor correlation[last term of (9)] is parallel to the known formula ot fluctuating hydrodynamics ot ordinary fluids /lo/. According to th? last term of (11) this eftect alone would yield the wrong sign wrong magnitude of rl - rlh above Th and the below
(ii)Thr
to
TA.
2 I$1 term in (3) models the Ipading dependence ot the ettec ive noncritical visb cosity ot superfluid He on the supertluid traction. Above Th this shows up as a local effect corresp ncilng to tluctuating clusters e of superfluid He, as eculated earl irr ill/. Below TA the ,$/“term contributes significantly already at the hydrodynamic (mean field) level.
The semiquantitative considerations Ferrell and collaborators /12.13/ do into account the effect (ii). Their based on the etfect (i) difter tram with respect to sign and magnitude. that their results provide a correct of the ettcctivr exponent ot q - oh
by not take results ours both We doubt explanation above Th.
*
c Is, 0
-*
-3 -A
I
I
I
-1 *-
-c=
-
g
-2
-
-3
-THEORY
log
Critical shear Data tram Rets. our expression
I
I
I
ItI
FTGURE 1 viscosity $= (Q - QA)/~A vs l-4. The solid lines represent (11) - (13).
t.
REFERENCES C. Howald, and H. Meyer, J. Low Ill S. Wang, Temp. Phys.. to be published. /2/ L. Bruschi, G. Mazzi, M. Santini. and G. Torzo, J. Low Temp. Phys. 29 (1977) 63. /3/ R. Biskeborn and R.W. Guernsey. Phys. Rev. Lett. 34 (1975) 455. /4/ R.W.H. Webeler and G. Allen, Phys. Rev. A 5 (1972) 1820. /5/ P.C. Hohenborg and P.C. Martin, Ann. Phys. (N.Y.) 34 (1965) 291. f6J J. Pankert and V. Dohm. Phys. Rev. B 40 (1989) 10842. 193. /7/ V. Dohm, 2. Phys. B 61 (1985) /8/ W.Y. Tam and G. Ahlers, Phys. Rev. B 33 (1986) 183: B 37 (1988) 7898. /9/ P.C. Martin, E.D. Siggia, and H.A. Rosr,, Phys. Rev. A 8 (1973) 423. /lO/L.D. Landau and E.M. Litschitz. Sov. Phys. JEPT 5 (1957) 512. /11/K. Mendelssohn. in Encyclopedia of Physics, Vol. XV, cd. S. Fliigge (Springer. Brri in, 1956). /12/R.A. Ferrell, J. Low Temp. Phys. 70 (1988) 435. /13/J.K. Bhattacharjer. R.A. Ferrell, and Z.Y. in Proceedings of 17th International Chen, Conierencc on Low Temperaturf, Physics, cds. LJ. Eckern, A. Schmid. W. Weber. and H. Wiihl (North-Holland, New York, 1984). p. 977.