Casimir effect in interacting Euclidean field theories - University of ...

Casimir effect in interacting
Euclidean field theories
H. W. Diehl
Fachbereich Physik, Universität Duisburg-Essen
Collaborators: Daniel Grüneberg, U. Duisburg-Essen
M. A. Shpot, ICMP Lviv (Ukraine)
Daniel Dantchev, BAS Sofia (Bulgaria)
Fred Hucht, U. Duisburg-Essen
Work supported in part by: DFG, grant # Die-378/5
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 1

Casimir effect in QED
July 15, 1909 – May 4, 2000
1948:
vacuum
fluctuations
L
vacuum
• normal modes of electromagnetic field between plates:
• ground-state energy:
E(L) = 1
2
q,µ
bound. cond. 1
conducting plates
bound. cond. 2
ωq = c |q|; q = (qx, qy, qz = m π/L), m ∈ N
ωq = CΛV + C s Λ A
“infinities”
− ∆ (1,2)
QED (d)
universal
• (fluctuation induced) force: FC(L) = − ∂E
∂L
c A
L
, ∆(D,D)
d QED
c
= −A d∆(1,2)
Ld+1 QED (d)
(3) = π2
720 .
reviews by: Bordag et al, Milton, Mostepanenko & Trunov, Elizalde & Romeo, . . .
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 2

Casimir effect in QED
July 15, 1909 – May 4, 2000
1948:
vacuum
fluctuations
L
vacuum
• normal modes of electromagnetic field between plates:
• ground-state energy:
E(L) = 1
2
q,µ
bound. cond. 1
conducting plates
bound. cond. 2
ωq = c |q|; q = (qx, qy, qz = m π/L), m ∈ N
ωq = CΛV + C s Λ A
“infinities”
− ∆ (1,2)
QED (d)
universal
• (fluctuation induced) force: FC(L) = − ∂E
∂L
c A
L
, ∆(D,D)
d QED
= − 0.013
(L/µm) 4
(3) = π2
720 .
dyn
A
cm2 reviews by: Bordag et al, Milton, Mostepanenko & Trunov, Elizalde & Romeo, . . .
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 2

Experimental Verification
S. Lamoreaux, PRL 87, 5 (1997);
U. Mohideen and A. Roy, PRL 81, 4549 (1998); parallel plates: G. Bressi et al, PRL 99, 041804 (2002)
polystyrene sphere (∅ 196 µm) and sapphire
plate coated with Au
plate-sphere separations from 0.1 to 0.9 µm
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 3

Experimental Verification
S. Lamoreaux, PRL 87, 5 (1997);
U. Mohideen and A. Roy, PRL 81, 4549 (1998); parallel plates: G. Bressi et al, PRL 99, 041804 (2002)
polystyrene sphere (∅ 196 µm) and sapphire
plate coated with Au
plate-sphere separations from 0.1 to 0.9 µm
solid line: Casimir force for plate-sphere geometry
including corrections due to
• finite conductivity
• surface roughness
• finite temperatures
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 3

Important Properties of Casimir Force (QED)
Casimir force (QED)
is independent of microscopic details (“universal”)
depends on gross features of
medium: space dimension d, dispersion relation, scalar / vector field,
geometry, . . .
boundaries: boundary conditions, geometry, curvature, . . .
usually is described by noninteracting (effective Gaussian) field theory
– coupling to matter field: only through boundary conditions
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 4

Important Properties of Casimir Force (QED)
Casimir force (QED)
is independent of microscopic details (“universal”)
depends on gross features of
medium: space dimension d, dispersion relation, scalar / vector field,
geometry, . . .
boundaries: boundary conditions, geometry, curvature, . . .
usually is described by noninteracting (effective Gaussian) field theory
– coupling to matter field: only through boundary conditions
Interacting Field Theories?
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 4

Important Properties of Casimir Force (QED)
Casimir force (QED)
is independent of microscopic details (“universal”)
depends on gross features of
medium: space dimension d, dispersion relation, scalar / vector field,
geometry, . . .
boundaries: boundary conditions, geometry, curvature, . . .
usually is described by noninteracting (effective Gaussian) field theory
– coupling to matter field: only through boundary conditions
Interacting Field Theories?
Yes, for condensed matter systems at critical points!
space dimension d < 4: Ginzburg criterion fails as T → Tc !
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 4

“Thermodynamic” Casimir Effect
pressure
0 0 K
solid
liquid
gas
critical
point
temperature
partition sum: Z =
FL,A(T)
kBT
φ
M.E. Fisher & P.-G. de Gennes (1978):
• large-λ modes ≈ massless
• consider confined nearly critical systems
B1 : area A
nearly critical fluid
B2 : area A
e −H[φ] = Dφ e −H[φ] = exp[−FL,A(T)/kBT]
= LA fbk(T) + A [fs,1(T, . . .) + fs,2(T, . . .)]
bulk contribution
surface contributions
L
+ A fx(T, L, . . .)
residual
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 5

“Thermodynamic” Casimir Effect
pressure
0 0 K
solid
liquid
gas
critical
point
temperature
FL,A(T)
kBT
= LA fbk(T)
bulk contribution
M.E. Fisher & P.-G. de Gennes (1978):
• large-λ modes ≈ massless
• consider confined nearly critical systems
B1 : area A
nearly critical fluid
B2 : area A
L
+ A [fs,1(T, . . .) + fs,2(T, . . .)]
surface contributions
Casimir force per area: FC(T, L, . ..)/A = −kBT ∂fx
∂L
finite size scaling (only short-range interactions):
fx(T, L, . . .) ≈ L −(d−1)
Y
universal
+ A fx(T, L, . . .)
residual
(L/ξ∞, ...) at Tc,∞ : fx ≈ ∆C
Casimir amplitude
L −(d−1)
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 5

What to expect?
1. only short-range interactions
At T = Tc,∞:
T > Tc,∞:
f sing
x
f sing
x
−(d−1)
≈ L ∆C(d)
L→∞
dominates
∆C, Y, . . . universal, dependent on
+ ∆ω,C(d) gω
Lω
+ . . .
Wegner corrections ω 0.8
−(d−1)
≈ L Y (L/ξ) +
L→∞ gω
Lω Yω(L/ξ)
+ ...
bulk universality class (d, n, short-range interactions)
gross features of both boundary planes
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 6

What to expect?
add long-range pair interaction ∝ b r −(d+σ)
12 : irrelevant when σ > 2 − η
2. with “irrelevant” long-range interactions
At T = Tc,∞:
f sing
x
−(d−1)
≈ L ∆C(d)
L→∞
dominates
T > Tc,∞:
f sing
x
+ ∆ω,C(d) gω(b)
Lω + ∆σ,C(d,σ)
Wegner corrections ω 0.8
−(d−1)
≈ L Y (L/ξ) +
L→∞ gω(b)
Lω Yω(L/ξ) +
∼
⎧
⎨e
−L/ξ , b = 0
⎩
b L −(d+σ−1) , b = 0
long-range contribution dominates asymptotically!
b
+ . ..
Lσ+η−2 b
Lσ+η−2 Yσ(L/ξ)
+ ...
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 6

Scaled Casimir Force
¨
©
1
10 -1
10 -2
10 -3
10 -4
10 -5
d = σ = 3, L = 50
(b) L = 50
b = 0
b = 2/3
asymptote
10
1 10 100
-6
¡¢ £
¤
¥ ¦§£
d + σ = 6
log corrections
D. Dantchev, H. W. Diehl, and D. Grüneberg: Phys. Rev. E 73, 016131 (2006)
periodic boundary conditions, exact spherical-model (n → ∞) result
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 7

Experimentall Verification: 4 He wetting films
mgh
gravitation
= γvdW
L 3
van der Waals
1
1 + L/L 1/2
retardation
copper plate
He vapor
liquid He
+ v kBT Ξ(L/ξ)
L 3
Casimir
L
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 8

Thinning of 4 He films near Tλ
Theory: in rather modest state
Krech & Dietrich 1991/92: T ≥ Tλ
Li & Kardar 1991: T ≪ Tλ (Goldstone modes)
Zandi, Rudnick & Kardar 2004: interface fluctuations
Experiment:
Garcia & Chan, PRL 83,
1187 (1999)
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 9

Thinning of 4 He films near Tλ
Theory: in rather modest state
Krech & Dietrich 1991/92: T ≥ Tλ
Li & Kardar 1991: T ≪ Tλ (Goldstone modes)
Zandi, Rudnick & Kardar 2004: interface fluctuations
Experiment:
Garcia & Chan, PRL 83,
1187 (1999)
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 9

Thinning of 4 He films near Tλ
Theory: in rather modest state
Krech & Dietrich 1991/92: T ≥ Tλ
Li & Kardar 1991: T ≪ Tλ (Goldstone modes)
Zandi, Rudnick & Kardar 2004: interface fluctuations
Experiment:
Garcia & Chan, PRL 83,
1187 (1999)
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 9

Monte Carlo Results
F. Hucht (U. Duisburg-Essen): cond-mat/0706.3458
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 10

Monte Carlo Results
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 10

Relevant Issues
bulk critical behavior at Tc,∞ as L → ∞
confined critical fluctuations, boundary field theory
finite size and boundary effects
pseudo-critical or critical behavior in slab at Tc(L) < Tc,∞ when L < ∞
dimensional crossover
low-T Casimir force from confined Goldstone modes
interface fluctuations
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 11

Relevant Issues
bulk critical behavior at Tc,∞ as L → ∞
confined critical fluctuations, boundary field theory
finite size and boundary effects
pseudo-critical or critical behavior in slab at Tc(L) < Tc,∞ when L < ∞
dimensional crossover
low-T Casimir force from confined Goldstone modes
interface fluctuations
From now on: T ≥ Tc,∞
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 11

RG and Bulk Critical Behavior
microscopic model, e.g. Ising model
H = −
Kij sisj − H
i=j
mesoscopic model: φ = order parameter field (|q| ≤ Λ)
H = d d
1
x
2 (∇φ)2 + ˚τ
2 φ2 + ˚u
4! φ4 −˚
h φ ,
behavior for q ≪ Λ: via renormalized field theory: Λ → ∞, requires renormalizations
dimensionless (renormalized) coupling constants {gj = τ, u,h, ...}
˚u = µ ɛ Zu(u,Λ) u , ˚τ −˚τc = Zτ µ 2 τ , ˚ h = µ (d+2)/2 Z −1/2
φ h , φ = Z 1/2
φ φR .
µ → µℓ ⇒ gj → ¯gj(ℓ) running interaction constants
ℓ d
dℓ ¯gj(ℓ)
= βj[¯g(ℓ)] , βj(¯g) = µ∂µ
j
sj
0 gj
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 12

2-Scale-Factor Universality
βu
d = 4 − ɛ < 4 ū(ℓ) ≈
ℓ→0 u ∗ + const (u − u ∗ ) ℓ ω
u
u ∗ = O(ɛ)
¯τ(ℓ) ≈
ℓ→0 Eτ(u)ℓ −
¯h(ℓ) ≈
ℓ→0
G(x, . . .;τ, h, u) ≈ ξ −dG−ηG EG(u)
powers of Eh, Eτ
universal
ν
τ
Eh(u) ℓ
nonuniversal
−∆/ν h
G(x/ξ, . . .;1, h ξ ∆/ν , u ∗ )
scaling function
universality (crit. exponents, scaling functions, amplitude ratios)
2-scale-factor universality
corrections to scaling from terms ∼ (u − u ∗ ) ξ −ω
, ξ ∼ τ −ν
τ =
T −Tc,∞
Tc,∞
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 13

L × ∞ d−1 Slabs
n-component φ 4 -model, V ≡ R d−1 × [0, L]
H[φ] =
V
d d x
↑
V
↓ ← periodic bc →
1
2 (∇φ)2 + ˚τ
2 φ2 + ˚u
4! φ4
antiperiodic bc: φ(x) = ±φ(x + L ˆz)
no new counter terms
L not renormalized ⇒ dependence on L/ξ !
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 14

L × ∞ d−1 Slabs
Tc,∞
n-component φ 4 -model, V ≡ R d−1 × [0, L]
B1
n
↑
V
↓
n
H[φ] =
V
d d x
1
2 (∇φ)2 + ˚τ
2 φ2 + ˚u
4! φ4
free surfaces: boundary condition: ∂nφ = ˚cj φ
T
surface & bulk
disordered SD/BD
SO/BD
ordinary special extraordinary
surface enhancement −c
SO/BO SO/BO
surface
Tc,s(c)
+
2
j=1
˚cj
2
B2
Bj
d d−1 r φ 2
ordinary: c = ∞, Dirichlet bc
φ(x) ≈ C(z)
z→0
∼z0.8 ∂nφ|B
stable fixed point!
special: c = 0 ⇔ ˚c = ˚csp = 0
φ ≈
z→0 D(z)
∼z −0.2
φ|B
Neumann bc not a fp!!
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 14

Previous Field Theory RG Results
RG analysis in d = 4 − ɛ dimensions:
boundary critical behavior: HWD & Dietrich, 80–; HWD 86, HWD 97
Casimir effect:
Symanzik 1981: ∆ (bc)
C (ɛ, n)/n = a0 + a1(n) ɛ
for Dirichlet-Dirichlet (D-D) boundary conditions
Krech & Dietrich 1991, 1992: ∆ (bc)
free bc
for bc = periodic, antiperiodic, D-D, D-sp, sp-sp
“sp” = “special” = “critically enhanced”
Claim: (HWD, Grüneberg & Shpot, 2006)
problems
theory ill-defined at Tc,∞ for some bc!
C (ɛ, n) and Y (bc) to O(ɛ)
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 15

Free Propagator
G (L)
bc (x; x′ ) =
periodic boundary conditions:
d d−1 p
(2π) d−1
m
〈z|m〉〈m|z ′ 〉
p 2 + k 2 m +˚τ eip·(r−r′ )
〈z|m〉 = L −1/2 e ikmz , km = 2πm/L , m = 0, ±1, ±2, . . .
antiperiodic boundary conditions:
Dirichlet-Dirichlet bc:
2
〈z|m〉 =
L sin(kmz) , km = πm/L , m = 1, 2, . . .
Dirichlet-Neumann bc:
Neumann-Neumann bc:
2 − δm,0
〈z|m〉 =
L
cos(kmz) , km = πm/L , m = 0, 1, 2, . . .
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 16

Free Propagator
G (L)
bc (x; x′ ) =
d d−1 p
(2π) d−1
m
〈z|m〉〈m|z ′ 〉
p 2 + k 2 m +˚τ eip·(r−r′ )
periodic boundary conditions: ∃ zero mode at ˚τ = 0 (T = Tc,∞)
〈z|m〉 = L −1/2 e ikmz , km = 2πm/L , m = 0, ±1, ±2, . . .
antiperiodic boundary conditions: no zero mode
Dirichlet-Dirichlet bc: no zero mode
2
〈z|m〉 =
L sin(kmz) , km = πm/L , m = 1, 2, . . .
Dirichlet-Neumann bc: no zero mode
Neumann-Neumann bc: ∃ zero mode at ˚τ = 0 (T = Tc,∞)
2 − δm,0
〈z|m〉 = cos(kmz) ,
L
km = πm/L , m = 0, 1, 2, . . .
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 16

Problems: for periodic and sp-sp bc
n dependence of scaling function
(periodic bc)
Θ(L/ξ∞)/n
0
-0.04
-0.08
-0.12
d = 3
n = 1
n = 2
n = 3
n→∞
n→∞ (exact)
-0.16
0 2 4 6 8
L/ξ∞
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 17

Problems: for periodic and sp-sp bc
n dependence of scaling function
(periodic bc)
Θ(L/ξ∞)/n
0
-0.04
-0.08
-0.12
d = 3
n = 1
n = 2
n = 3
n→∞
n→∞ (exact)
-0.16
0 2 4 6 8
L/ξ∞
⇒ theory ill-defined beyond 2 loops
periodic and sp-sp boundary conditions involve zero modes at Tc,∞!
♥ ♥♥ = infrared singular at Tc,∞
violation of analyticity requirements at Tc,∞ when L < ∞
can be remedied: HWD, D. Grüneberg & M.A. Shpot: EPL 75, 241 (2006)
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 17

Revised Field Theory
φ(r,z) = ϕ(r)
0-mode contribution
+ ψ(r, z) ,
L
0
dz ψ(r,z) = 0 .
effective (d − 1)-dimensional FT: e −H eff[ϕ] ≡ Trψ e −H[ϕ+ψ]
ϕ ϕ gives shift ˚τ → ˚τ (L)
bc
Heff[ϕ] = Fψ + H[ϕ] − ln e −H int[ϕ,ψ]
= ˚τ + δ˚τ(L)
bc
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 18

Revised Field Theory
φ(r,z) = ϕ(r)
0-mode contribution
+ ψ(r, z) ,
L
0
dz ψ(r,z) = 0 .
effective (d − 1)-dimensional FT: e −H eff[ϕ] ≡ Trψ e −H[ϕ+ψ]
ϕ ϕ gives shift ˚τ → ˚τ (L)
bc
F = Fψ + ∼(u ∗ ) (3−ɛ)/2
Heff[ϕ] = Fψ + H[ϕ] − ln e −H int[ϕ,ψ]
= ˚τ + δ˚τ(L)
bc
+ + . . .
RG improved perturbation theory infrared well-behaved at T = Tc,∞ !
. . . not at Tc,L ⇒ must require T ≥ Tc,∞ (as for nonzero-mode bc)
⇒ ɛ 3/2 , ɛ 5/2 , ɛ 5/2 ln ɛ, ... contributions to Casimir amplitudes
(in conformity with exact n → ∞ solution)
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 18

extrapolated ɛ-expansion results:
periodic boundary conditions
¦
§
¥
¤
¨
©
0
-0.05
-0.10
-0.15
n = 1 (Ising), d = 3
-0.20
0 2 4 6 8
0
-0.05
-0.10
-0.15
-0.20
n = 3 (Heisenberg), d = 3
-0.25
0 2 4 6 8
¡¢ £
0
-0.05
-0.10
-0.15
-0.20
0
-0.10
-0.20
-0.30
-0.40
n = 2 (XY), d = 3
0 2 4 6 8
n = ∞ (Spherical), d = 3
-0.50
0 2 4 6 8
Krech &
Dietrich 91
Grüneberg
& HWD:
to be published
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 19

Summary / Conclusions
Condensed matter systems:
rich lab to study fluctuation induced (“Casimir”) forces
involve interacting or noninteracting field theories
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 20

Summary / Conclusions
Condensed matter systems:
rich lab to study fluctuation induced (“Casimir”) forces
involve interacting or noninteracting field theories
good interaction between theory and experiment
now also with Monte Carlo simulations
encouraging progress
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 20

Summary / Conclusions
Condensed matter systems:
rich lab to study fluctuation induced (“Casimir”) forces
involve interacting or noninteracting field theories
good interaction between theory and experiment
now also with Monte Carlo simulations
encouraging progress
but challenging problems remain,
e.g., dimensional crossover
inclusion of proper low-T behavior
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 20

Summary / Conclusions
Condensed matter systems:
rich lab to study fluctuation induced (“Casimir”) forces
involve interacting or noninteracting field theories
good interaction between theory and experiment
now also with Monte Carlo simulations
encouraging progress
but challenging problems remain,
e.g., dimensional crossover
inclusion of proper low-T behavior
Thank You!
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 20

Further Reading
References
1. D. Dantchev, H. W. Diehl & D. Grüneberg: Phys. Rev. B 73, 016131 (2006)
2. H. W. Diehl and D. Grüneberg and M. A. Shpot: EPL 75, 241 (2006)
3. D. Grüneberg and H. W. Diehl: to be published
4. F. Hucht: cond-mat 0706.3458
Background
Boundary critical phenomena
a) H. W. Diehl: “Field–theoretical Approach to Critical Behaviour at
Surfaces”, in: Phase Transitions and Critical Phenomena, edited by C.
Domb and J. L. Lebowitz (Academic, London, 1986), Vol. 10, pp. 75–267.
b) H. W. Diehl, Int. J. Mod. Phys. B 11, 3503 (1997), cond-mat/9610143.
Thermodynamic Casimir effect
c) M. Krech, Casimir Effect in Critical Systems (World Scientific, 1994)
HW Diehl, QTS5, Valladolid, July 22–28, 2007 – p. 21