Oscillating Casimir force between two slabs in a Fermi sea

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1 Osillting Csimir fore etween two sls in Fermi se Chen Li-Wei( ) ), Su Guo-Zhen( ) ), Chen Jin-Cn( ) ), nd Andresen Bjrne ) ) Deprtment of Physis nd Institute of Theoretil Physis nd Astrophysis, Ximen University, Ximen 3615, Chin ) Niels Bohr Institute, University of Copenhgen, Universitetsprken 5, DK-1 Copenhgen Ø, Denmrk (Reeived 8 June 11; revised mnusript reeived 16 August 11) The Csimir effet for two prllel sls immersed in n idel Fermi se is investigted t oth zero nd nonzero tempertures. It is found tht the Csimir effet in Fermi gs is distintly different from tht in n eletromgneti field or mssive Bose gs. In ontrst to the fmilir result tht the Csimir fore dereses monotonilly with the inrese of the seprtion L etween two sls in n eletromgneti field nd mssive Bose gs, the Csimir fore in Fermi gs osilltes s funtion of L. The Csimir fore n e either ttrtive or repulsive, depending sensitively on the mgnitude of L. In ddition, it is found tht the mplitude of the Csimir fore in Fermi gs dereses with the inrese of the temperture, whih lso is ontrry to the se in Bose gs, sine the osoni Csimir fore inreses linerly with the inrese of the temperture in the region T < T, where T is the ritil temperture of the Bose Einstein ondenstion. Keywords: Csimir effet, Fermi gs, prllel sl, osilltion PACS: 5.3. d, 3.75.Ss, 5.3.Fk DOI: 1.188/16746/1/1/ Introdution The Csimir effet is n ttrtive intertion etween two losely sped prllel pltes used y the vuum flutution of the eletromgneti field. Aout 1 yers fter its predition in 1948 y Csimir, 1] the Csimir fore ws mesured experimentlly y Sprny. ] More urte mesurement of the Csimir fore ws performed in series of modern experiments 3 7] strting with Lmoreux s lndmrk experiment in ] In theoretil developments, gret del of effort hs een devoted to the lultion of the Csimir energy or the Csimir fore for different geometries nd different oundry onditions, 8 13] inluding those rel strutures of the oundries. Its importne for prtil pplitions is now eoming more widely ppreited in quntum field theory, Bose Einstein ondenstes, tomi nd moleulr physis, grvittion nd osmology, nd mthemtil physis. Generlly, the presene of oundries inside ny wve field n use Csimir-like effets if the ounded spe is smller thn the mximum wvelength of the wve field. For exmple, n ousti Csimir fore etween two prllel rigid pltes due to the rdition pressure of the nd-limited ousti noise ws explored oth theoretilly nd experimentlly. 14] Similrly, it my e expeted tht Csimir-like effets my our in quntum gs euse of the wve hrter of the gs toms. In Ref. 15], the Csimir fore etween two sls immersed in perfet Bose gs ws lulted for vrious oundry onditions. It ws found tht the Csimir fore hs the stndrd symptoti form with universl Csimir terms elow the ulk ritil temperture of Bose Einstein ondenstion (BEC) T nd vnishes exponentilly ove T. A question nturlly omes to mind: wht hppens if the Bose gs is repled y Fermi gs? The Csimir effet in fermioni fields hs een extensively investigted in the literture. In this onnetion, lrge mount of work hs een devoted to the lultion of the Csimir intertion etween two impurities or two ules immersed in Fermi se, 16 18] whih is prtiulrly relevnt to the physis of neutron strs 19] nd qurk gluon plsms. ] In most of those investigtions, the Csimir fore ws lulted utilizing the geometry-dependent density of sttes, 16,17] Projet supported y the Ntionl Nturl Siene Foundtion of Chin (Grnt No ) nd the Nturl Siene Foundtion of Fujin Provine, Chin (Grnt No. A1116). Corresponding uthor. E-mil: gzsu@xmu.edu.n 1 Chinese Physil Soiety nd IOP Pulishing Ltd

2 in whih the orretions rising from the presene of ostles hd een tken into ount. As opposed to those previous studies, ext numeril lultions will e employed in the present pper to lulte the Csimir fore etween two sls immersed in Fermi se.. Fermioni Csimir effet t zero temperture We onsider two lrge prllel sls immersed in se of perfet Fermi gs. The two sls re seprted long the z xis y smll distne L A, where A is the re of the sl. The system of the fermions etween the two sls (onfined system) is in thermodynmi equilirium with the Fermi se outside the sls (surroundings). The single-prtile energy of the onfined system is given y ε(k) = m (k x + k y + k z), (1) where m is the mss of the prtile, is the redued Plnk onstnt nd k is the wve vetor. For the prolem under onsidertion, k x nd k y re tken to e ontinuous, nd k z is quntized s k z = πn L, () with n = 1,, 3,... for the Dirihlet oundry ondition (DBC), n =, 1,,... for the Neumnn oundry ondition (NBC), nd n =, ±, ±4,... for the periodi oundry ondition (PBC). We first onsider the se t zero temperture (T = K). The thermodynmi potentil of the idel Fermi gs t K is given y Ω = k nd n e derived s ε F ε(k)]θ (ε F ε(k)), (3) Ak F ε F 8π Ω = Ak F ε F 8π J 1, J ( πn 1 k F L 1 ( πn k F L ) ] ) ], for PBC, (4) where k B is the Boltzmn onstnt, ε F is the Fermi energy of the onfined system, whih is equl to tht of the surroundings in the stte of equilirium, k F = mε F /, J 1 nd J re the integer prts of k F L/π nd k F L/(π), respetively, nd the summtion J 1, strts from 1 for DBC nd for NBC. The degenery relted to the internl struture of the prtiles is ssumed to e one for simpliity. From Eq. (4), the Csimir fore per unit re n e derived s ( Ω P C (L) = 1 ) ( ) ] Ω lim A L L ε F,A L ε F,A { ( ) πε F J 1 ] πn L 3 n 1 ( ) } 3, k F L 15 π = { ( ) πε J F ] πn L 3 (n) 1 ( ) } 3 (5), for PBC. k F L 15 π Bsed on Eq. (5), we n understnd the dependene of the Csimir fore on the seprtion of the two sls. For exmple, in the se of DBC, the sled Csimir pressure n e diretly derived from Eq. (5) to e P C = 5 J 1 (J 1 + 1/)(J 1 + 1) P S (k F L/π) 3 1 3J 1 + 3J 1 1 5(k F L/π) ] 1, (6) where P S is the pressure of the surroundings t K nd is given y P S = ε F kf 3 /(15π ). 1] In Eq. (6), the numertors ontin J 1 nd thus remin onstnts over eh integer intervl, while the denomintors ontin (k F L/π) 3 nd (k F L/π) nd inrese s k F L/π inreses. For k F L/π < 1, J 1 = nd P C /P S = 1, whih mens tht the Csimir fore per unit re is equl to the pressure of the surroundings. This re- 151-

3 sult is expeted. When k F L/π < 1, the energy of the lowest level of the onfined system ε = π /(ml ) is lrger thn the Fermi energy ε F = kf /(m), so there will e no prtile nd hene no pressure in the spe etween the two sls. For k F L/π 1, P C /P S hs minimum P C, min P S = 5J 1 1 4J 4 1 t k F L/π = J 1 (J 1 = 1,, 3,... ) nd mximum (7) P C, mx P S = J 1(J 1 + 1/)(J 1 + 1) (J 1 + J 1 1/3) 3/ 1 (8) t k F L/π = J 1 + J 1 1/3 in eh integer intervl from J 1 to J Thus, P C /P S s funtion of k F L/π will osillte with the period of (k F L/π) = 1, s shown in Fig. 1. The property is not seen in the se of n eletromgneti field or mssive Bose gs, where the Csimir fore is known to e monotonilly deresing funtion of L. In eh integer intervl J 1 k F L/π < J 1 + 1, P C /P S my vry etween P C, min /P S < nd P C, mx /P S >. It indites tht the Csimir fore n e ttrtive or repulsive, lternting periodilly with the inrese of the seprtion etween the two sls. This gives rise to nother importnt differene in the Csimir effets etween the Bose nd the Fermi gses, s the Csimir fore for the former n only e ttrtive. 15] In the ses of NBC nd PBC, the dependene of the Csimir fore on the seprtion etween the two sls n e similrly disussed. It is found tht the urve of P C /P S versus k F L/π in the se of NBC is the sme s tht in the se of DBC, while in the se of PBC, P C /P S osilltes s funtion of k F L/π with douled period, s shown in Fig. 1. PC PS PC PS DBC nd NBC PBC DBC nd NBC PBC Fig. 1. Csimir fores versus seprtion etween the two ounding sls for different oundry onditions t zero temperture. Pnels nd give the results t smll nd lrge seprtions, respetively. It is importnt to note tht the prtile density of the onfined system is generlly different from tht of the surroundings when they re in thermodynmi equilirium t onstnt temperture nd hemil potentil. Aording to Eq. (4), the differene in prtile density etween the onfined system nd the surroundings n e otined s ρ(l) = 1 ( ) ( ) ] 1 Ω 1 Ω lim A L ε F L A,L L ε F A,L { k ( ) ] J 1 F πn 1 ( ) } 4πL, k F L 3 π = { kf ( ) ] J πn 1 ( ) }, for PBC. 4πL k F L 3 π (9) By using Eq. (9), the urves of ρ/ρ S versus k F L/π for different oundry onditions n e plotted, s shown in Fig., where ρ S is the prtile density of the surroundings t K nd is given y ρ S = kf 3 /(6π ). 1] It is found tht the urves eh disply swtooth-like osilltion tht eomes smoother s L inreses. The vlues of ρ/ρ S re sensitive to the oundry onditions: ρ/ρ S is negtive for DBC, positive for NBC, nd vries lterntely from positive to negtive for PBC. When L, ρ/ρ S = 1 for DBC, nd ρ/ρ S for NBC nd PBC

4 Dρ ρs Dρ ρs : DBC.4 : NBC : PBC -.4 : DBC : NBC : PBC Fig.. Differene etween the density of fermions etween the two sls nd tht outside the two sls versus seprtion of the sls for different oundry onditions t zero temperture. Pnels nd give the results t smll nd lrge seprtions, respetively. 3. Fermioni Csimir effet t nonzero tempertures nd ρ(l, T ) = k ( T πn f 1 4πL k T L ] π1/ k T L π f 3/(z) ) ]), (13) respetively, where the fugity z is determined y f 3/ (z) = ( TF T ) 3/ 4, (14) 3π1/ nd T F = ε F /k B is the Fermi temperture. The dependenes of P C (L, T ) nd ρ(l, T ) on prmeters L nd T n e otined from Eqs. (1) (14) using numeril lultion. Figure 3 shows the urves of P C /P S vrying with k F L/π for different vlues of the sled temperture T/T F. It n e seen tht the mplitude of P C /P S dereses with the inresing temperture. This property is different from tht for the Bose gs, in whih the Csimir fore inreses linerly with the inrese of the temperture in the region T < T. 15] Compred with the se T = K, the urves of P C /P S versus k F L/π t nonzero tempertures eome smooth due to therml exittion. We now turn to the ses of nonzero tempertures nd first onsider the se of the Dirihlet oundry ondition, for whih the thermodynmi potentil of the onfined system t nonzero tempertures n e expressed s Ω = k B T k = Ak T k BT 4π ln1 + z e βε(k) ] ( ) ]) πn f, (1) k T L 1 PC PS 1 PC PS where k T = mk B T /, z = e µ/(kbt ) is the fugity, µ is the hemil potentil of the onfined system, whih is equl to tht of the surroundings in the stte of equilirium, f ν (x) is the Fermi integrl f ν (x) = 1 t ν 1 dt Γ(ν) x 1 e t + 1, (11) nd Γ(x) is the Gmm funtion. Aording to Eq. (1), we n find the Csimir fore per unit re nd the differene in prtile density etween the onfined system nd the surroundings to e P C (L, T ) = πk BT L 3 π1/ 4 ( ) ]) πn n f 1 k T L ( ) 3 kt L f 5/(z)] (1) π 1 PC PS () Fig. 3. Csimir fore versus seprtion etween the two sls t T/T F =, T/T F =., nd () T/T F =.4. Figure 4 shows the urves of P C /P S vrying with T/T F for different vlues of prmeter k F L/π. It is oserved tht the urves n e signifintly hnged for slight vrition of k F L/π. It indites tht the Csimir fore is quite sensitive to the seprtion etween the sls. 151

5 1 3 PC PS f e d. /. / 1. / d. / 3 e. / 4 f. / T T F Fig. 4. Csimir fore s funtion of temperture for different seprtions etween the sls. 1 Dρ ρs 1 Dρ ρs 1 Dρ ρs () Fig. 5. Differene etween the density of fermions etween the two sls nd tht outside the two sls s funtion of the seprtion of the sls t T/T F =, T/T F =., nd () T/T F =.4. Figure 5 gives the urves of ρ/ρ S versus k F L/π for different vlues of prmeter T/T F, whih re otined using Eqs. (13) nd (14). It n e seen tht the osilltion of ρ/ρ S with k F L/π grdully disppers with the inrese of the temperture. In ddition, it is found tht for given vlue of k F L/π, the vlues of ρ/ρ S re out the sme for different vlues of T/T F. This implies tht the differene in prtile density etween the onfined system nd the surroundings is insensitive to the temperture in the low-temperture region. The fermioni Csimir effets t nonzero tempertures re quite similr for the ses of NBC nd PBC. Thus, they re not listed in detil here. 4. Conlusion We study the Csimir effet for two prllel sls immersed in se of perfet fermions. Some importnt results otined re s follows. (i) The Csimir fore in the idel Fermi gs osilltes with the inrese of the seprtion etween the two sls, whih is distintly different from tht in the se of n eletromgneti field or mssive Bose gs. (ii) The fermioni Csimir fore n e either ttrtive or repulsive, whih is nother importnt differene in the Csimir effet etween the Bose nd the Fermi gses. (iii) The mplitude of the Csimir fore dereses with the inrese of the temperture. This is opposite to tht for the Bose gs, in whih the Csimir fore inreses linerly with the inrese of the temperture in the region T < T. (iv) The differene in prtile density etween the onfined system nd the surroundings is sensitive to the seprtion etween the two sls nd the oundry onditions ut is insensitive to the temperture in the low-temperture region. Referenes 1] Csimir H B G 1948 Pro. Ned. Akd. Wet. B ] Sprny M J 1958 Physi ] Lmoreux S K 1997 Phys. Rev. Lett ] Mohideen U nd Roy A 1998 Phys. Rev. Lett ] Bressi G, Crugno G, Onofrio R nd Ruoso G Phys. Rev. Lett ] Antonini P, Bressi G, Crugno G, Glezzi G, Messineo G nd Ruoso G 6 New J. Phys ] Petrov V, Petrov M, Bryksin V, Petter J nd Tshudi T J 7 Exp. Theor. Phys ] Csdio R, Gruppuso A, Hrms B nd Miu O 7 Phys. Rev. D ] Zhi X H nd Li X Z 7 Phys. Rev. D ] Alves D T nd Grnhen E R 8 Phys. Rev. A ] Lim S C nd Teo L P 9 New J. Phys ] Bi Z W 4 At Phys. Sin (in Chinese) 13] Zeng R, Yng Y P nd Liu S T 8 At Phys. Sin (in Chinese) 14] Lrrz B nd Denrdo B 1998 Phys. Lett. A ] Mrtin P A nd Zgrenov V A 6 Europhys. Lett ] Bulg A nd Wirz A 1 Phys. Rev. Lett ] Wirz A, Bulg A nd Mgierski P 6 J. Phys. A: Mth. Gen ] Reti A, Fuhs J N, Peç C S nd Zwerger W 5 Phys. Rev. A ] Bulg A nd Mgierski P 1 Nul. Phys. A ] Neergrd G nd Mdsen J Phys. Rev. D ] Pthri R K 197 Sttistil Mehnis (nd edn.) (Oxford: Pergmon Press) p