Cri$cal Casimir forces from the equa$on of state of quantum cri$cal systems

Transcription

1 Cri$cal Casimir forces from the equa$on of state of quantum cri$cal systems A. Rançon D. Lopes Cardozo T. Roscilde P. Holdsworth Ecole Normale Superieure de Lyon F. Rose N. Dupuis LPTMC Paris 6 L.-P. Henry Innsbruck ERG 216

2 Introduc$on Casimir 1948 : Casimir effect a

3 Cri$cal Casimir force Fisher and de Gennes 1978 : fluctua$ng (classical) medium between two plates creates a force (if correla$on length of the order of distance between the plates) Large correla$on length : second order phase transi$on Also implies universality of the force close to cri$cality : / T T c Pressure of fluid on the plates : P = F = aa(f bulk + f ex ) f ex = a 3 fs (a/ ) P c a (a f s )=a 3 #(a/ )

4 Experimental realiza$ons Helium films Binary mixtures MC simula$ons : Vasilyev et al. 27 He experiments : Garcia et al Ganshin et al. 26 Binary mixture : Fukuto et al. 25 Strict boundary condi$ons

5 Boundary condi$ons Casimir scaling func$on universal, depends on : - dimension - symmetry of order parameter - geometry / boundary condi$ons ϑp (x) XY periodic BC (iii) L=1 L=15 L=2 Ref.[16], L=2 Ref.[21] x Vasilyev et al. 29

6 From cri$cal Casimir to quantum cri$cal systems Cri$cal classical Casimir force XY periodic BC.4.3 "Thermal part" of pressure of a 2D Bose gas ϑp (x) (iii) L=1 L=15 L=2 Ref.[16], L=2 Ref.[21] x VS Func$onal renormaliza$on group study : AR, Kodio, Dupuis, Lecheminant (213)

7 Quantum phase transi$ons QPT : transi$on at zero temperature (change of ground-state) when changing non-thermal parameter Examples : Bosonic Mog transi$on at constant density (XY universality class) Superfluid Mog insulator interac$on Ferro-paramagne$c transi$on in quantum Ising model in transverse field Ferromagnet paramagnet transverse field

8 Phase diagram and cri$cal scaling T Quantum cri$cal regime superfluid QCP Mog insulator interac$on Close to QCP : f(,t)= ( )+T 3 /c 2 fs ( c/ ) Ground-state energy density / c

9 Phase diagram and cri$cal scaling Thermal part of pressure/free energy SF QCR.1 Mog Close to QCP : f(,t)= ( )+T 3 /c 2 fs ( c/ )

10 Quantum classical correspondence L? a PBC critical fluctuations (a) periodic confinement (b) imaginary $me L L confined 3D classical system a=c/t 2D quantum system at finite T Close to a cri$cal point, universality implies that scaling func$ons are the same! See e.g. Sachdev

11 From cri$cal Casimir to quantum cri$cal systems and back f(,t)= ( )+T 3 /c 2 fs ( c/ ) Universality : same as confined 3D classical system = L 2 hĥi ( f) Average energy of a quantum critical system : = T 3 c 2 # P ( c/ ) Thermodynamic stability of quantum systems implies agrac$ve Casimir force for periodic BC. P c a (a f s )=a 3 #(a/ ) For free bosons : Coleman et al. Am. J. Phys. 29

12 Cri$cal Casimir vs Equa$on of State L? a PBC critical fluctuations (a) periodic confinement (b) imaginary $me L L Quantum to classical correspondence : average energy interpreted as (universal) entropic "Casimir" force Classical to quantum correspondence : cri$cal Casimir force with PBC can be quantum simulated experimentally!

13 FRG calcula$on Theore$cal approaches are scarce to compute scaling func$ons (large N or epsilon expansion fail here). AR, Kodio, Lecheminant and Dupuis (214) : LPA'+ expansion up to φ 4 Not good for Ising. Improved calcula$on : 2 nd order of Deriva$ve Expansion k[ ]= Z ~ d Z + Y x k ( ) 4 Z d d x r k ( ) 2 (r ) 2 + Z k ( ) 2 (@ ) 2 (r ) 2 + Y k ( ) (@ ) 2 + U k ( ), (5) 4 t = 2 /2. f =lim k! U k (,k ) See also : Jakubczyk and Napiorkowski 213

14 Casimir cri$cal force from the FRG TABLE II. Universal Casimir amplitude #(, #()/2 )/2. N NPRG Monte Carlo [5].152(2).2993(7) Ising universality class XY universality class ϑ(x, ) -.2 # P (x) #(x) x x MC : Vasilyev et al. 29 Hucht et al. 211 D. Lopes Cardoso (PhD thesis 215) x =(a/ ) 1/ (T T c ) x x MC : Vasilyev et al. 29

15 Scaling func$ons for different N -.1 ϑ/n sg(x) x ν N =1 N =2 N =3 large N

16 Conclusion and perspec$ves - Cri$cal Casimir forces with periodic BC is the equa$on of state of a quantum cri$cal system. - Corresponding scaling func$ons could be measured in state of the art experiments on quantum systems. - Tools of quantum many-body problem can be used to study cri$cal Casimir forces (Quantum Monte Carlo). - Open ques$on : how to tackle other boundary condi$ons? Ex : Free BC, order parameter depends on posi$on, flow equa$on much harder to solve. arxiv:

17 Phase diagram : quantum vs classical T QCP δ 1/a bulk limit T c T

18 Finite size scaling for quantum systems QMC: β>>l β L Cri$cal Casimir : β<<l L β L L We can thus expect that the universal coefficient of FSS depends on the ra$o ρ=βc/l. T 3 At the cri$cal point δ=: u = E ( ) () c 2 Casimir amplitude (=-.32 for Ising) lim!1 ( ) = 3 with α a universal (non-standard Casimir) amplitude (=.37 for Ising)

19 Aspect ra$o and Finite Size Scaling Dependence on aspect ra$o known in context of Casimir forces. Con$nuous imaginary $me QMC for quantum Ising in transverse field.4.2 QMC MC y=β/l=a/l -.2 MC: Hucht et al y

20 Non-Pertuba$ve Renormaliza$on Group Subtle calcula$on in 4-ε and Large N for periodic BC. Here : Non-Perturba$ve Renormaliza$on Group (Wegerich 1993) T8@+9;&AU&8,)+AS5&+SF(V(F&Q;&@A@(S)P@&5,89(&W&!"#! $!"#! " k k [ ] &&5;5)(@&!!!#$!%&'()*+,-!& " ' " (). #)*!"#! "%" %! X;5)(@&AU& &&+S)(*(5) S k [ ]= X i Z q R k (q) i (q) i ( q) # $ '%) $ # $ %

21 E odel a famh that is low- In the thisis portional to the thermal energy, which is always positive. model [27], the The situation for general y is discussed further below. A quantum-to-clas summary of the conversion Effec$ve fromac$on the classical to the quantum terminology is given in Table I. from a derivative results, which ar ( Gibbs free energy) Renormalization group calculation of the critical ond order and im Casimir force in O(N) models - The two-dimensional the Supplementa quantum O(N) model Quantum is defined O(N) bymodel the action Figure 2 show Z ~ Z (r') S = d d 2 2 r + (@ ') 2 2 2c 2 + r'2 2 + u('2 ) 2 tained from the, within 6 the NPR 4! Ising (N=1), X (9) versality classes, Effec$ve powers ac$on of: Legendre q 2 i /k2 and transform! n 2 i /(ck) of Free 2 energy when with q i respect,! ni /c to magne$c k. field. The derivative expansion of the e ective action is fully determined by the O(N) symmetry = h'i of the model. To second order, Depends on the order parameter : Ansatz : Deriva$ve expansion (low energy fluctua$ons most important close to QCP) k[ ]= Z ~ d Z + Y x k ( ) 4 Z d d x r k ( ) 2 (r ) 2 + Z k ( ) 2 (@ ) 2 (r ) 2 + Y k ( ) (@ ) 2 + U k ( ), (5) 4 where we have introduced the O(N) invariant = 2 /2. = 2 /2