- Email: [email protected]
Some new aspects of RVB theory in high-Tc cuprates
PHYSICAi
Phyfica B 186-188 (1993) 808-812 NortHland
Some new aspects of R V B theory in high-T c cuprates Naoto Nagaosa D ~ t m e m of App~d P h y ~ , Un~ersi~ of Tokyo, Hongo, B u n ~ k u , Tokyo 113, Japan We discuss some new aspec~ of RVB theory. By ta~ng into account the fluctuations of the spinon pMfing and the holon condensation, the crossover phenomena in the normM phase of high-T~ superconductors are clarified and compared with recent experiments.
I. Introduction
It is now well estabfished that the anom~ous properties of high-T, s u p e r c o n d u ~ o ~ are a~fibuted to strong electron correlations [1]. One of the most impo~ant t h e o r e t ~ issue is the posfibility of the breakdown of the Fermi hquid theory in this high~ correlated d e ~ r o n sy~em. The resonating v~ence bond (RVB) state first proposed by Anderson [2] shows non-Fermi liquid b e h a ~ o r where s ~ n and charge degrees of ~ e e d o m are somehow separated, and carried by two kinds of quaff particle, i.e. spinon and holon, respective~. The mean field theory for this spin~harge separated sy~em predicts an intere~ing phase diagram wh~h is schematically shown in fig. l(a) [3]. In r e , o n I, neRher the spinon p~fing nor the holon condensation occur. The n o r m ~ state properties of the "~range m e t ~ state" has been ~udied taking into account the gauge field fluctuations. In r e , o n II, only holon condensation occurs wh~e the spinon rem ~ n s in the n o r m ~ state. This state is a Fermi liquid fimHar to the heavy fermion. In r e , o n III, only the spinon p~fing occurs while holon rem~ns normS. This ~ cal~d the "spin gap state" because the gap in the spin exaltation is introduced due to p~fing fim~ar to the BCS state. Tanamoto et ~ . [4] have recent~ investigated the magnetic properties of the spinons in this spin gap state as well as the s~ange metal state and concluded that the NMR and neutron sc~tefing data can be expl~ned by the spinon with the dispe~ fion ~ v e n by the mean field theory if exchange enhancement ~ taken into account. However, spinons and holons are not independent of each other. They
Correspondence to: N. Nagaosa, Depa~ment of Applied Phy~cs, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan.
are c o u p ~ d through the local c o n f r o n t that the sum of the spinon number and holon number should be equal to 1 at every ~te. This c o n f r o n t ~ conveniently expressed in terms of the gauge field, and the meaning of the spin-charge separation has been investigated in
s~r~.gomota~ J
(~)
~
/b)
TD
~ ~ d
"\,
//
\'\,,/../" .~. ~
~
TBE
) g Fig. 1. (a)The mean-fidd phase diagram. (b) Modified phase diagram when in~astic fifetime effe~s are taken into account. The only true phase ~ansition ~ across the heavy line into the supercondu~ing phase.
0921-4526/93/$06~0 © 1993 - Ege~er Sdence Pubfishers B.V. All fights reserved
~ Na~a
/ New a ~
~ r m s of gauge fie~ theory ~ - ~ . In t~s language, the s~n~harge separation co~esponds to the deconfi~ng phase while the Fermi fiq~d co~esponds to the confin~g phase. At f i ~ ~ m p e r ~ u r e , lhe coupling with the gauge fie~ has been taken ~ t o account by the G~gzburg-Landau (GL) theory ~ d u d ~ g the two order p a r a m e ~ , i.e., those of the spinon p~ring and h ~ o n condensation [~. One condufion is t h ~ the mean field ~ansition of the s~non p~fing ( h o r n condensation) without h o r n condensation ~ n o n p~ring) becomes a crossover and the true phase tranfition is lhe supe~onducting one ~ wh~h both the order param~ers become nonzero fimuRaneou~y as shown in fig. l(b). ARhough there ~ no phase ~anfition in the norm~ phase, R is impo~ant to study the change ~ the physic~ properties across these crossovers because sever~ experiment~ resuRs show ~teresting beha~or as the concen~ation is changed. In t~s paper, we focus on the Uanspo~ properties in the n o r m ~ phase. The pre~ous an~ysis of the ~range metal r e , o n is extended to the s~n gap as well as the Fermi l ~ d r e , o n . The ~anspo~ properties ~ong the c-a~s are ~so ~scu~ed as well as the ~-plane properties.
2. Summary of some experimental results
Before discussing the theoretical results, we ~ummarize the relevant expefiment~ resuks recently obtained by Uchida et ~. [9]. They have measured: (1) the resistivity p~o in the plane; and (2) p~ ~ong the c-axis; (3) the H~I con~ant in the plane (with the magnetic field perpendicular to the plane); (4) the thermopower S,o parallel to the plane; (5) S~ ~ong the c-ax~ cf the LSCO tingle ~ r y ~ chan~ng the hole concentration. The resul~ are: (1) When the hole concentration x is s m e a r than the optim~ (highe~ Tc) concen~ation Xm=0.15 , ~ b begins to deviate from T-finear to the lower v~ue as the temperature is lowered below T = 150K (for x = 0.1). For x = Xm, the T-hnear beha¼or ~ observed down to the superconducting To. When x exceeds x m appredably, the refistivity has a pofitive curvature, i.e., Pab ~ T~ (a = 1.5). (2) The behavior of p~ changes dramatic~ly as the concentration x increases. In the low x region, the temperature dependence h insulating and the absolute value is much larger than the MoR fimit. As x ~ increased, the temperature dependence becomes weaker and weaker, and the absolute v~ue decreases and approaches the Mo~ hmit. In the overdoped region, the temperature dependence ~ met~hc and the absolute v~ue ~ less than the Mo~ hmR. Rice [10]
~ RVB ~
~ h ~ h - ~ cupmtes
~9
has pointed out the possible co~espondence of the insulating beha¼or of the re~stifity and the e ~ e n c e of the pseudo gap in the magnet~ exaltations. (3) The H~I constant decreases as the concentration is increased. It is positive for x < 0.3 and turns negative for x > 0.3. In the low concentration r e , o n , the absolute v~ue ~ large and has a peak above the superconducting T~. Near the optimal concentration x m, it ~ continuously decreasing with temperature. For x > 0.3, the Hall constant ~ negative and ~most temperature independent. (4) Thermopower paralld to the plane ~ positive with the absolute value ~ O . l k B / e . There ~ ~so a report that S.~ is much less sen~tNe to a large magnetic field than that expe~ed from the spin splitting effect. (5) Thermopower S~ ~ong the c-a~s ~ ~so positive and approximate~ expre~ed as S~ ~ ( k B / e ) ( T / 2 0 0 K) for x =0.15. These expefiment~ features seem to violate the Fermi fiquid pi~ure. P a ~ u l a d y Anderson d ~ m e d that the met~l~ behafior within the plane and the insulating behafior between the plane are not compatib~ with each other within the Fermi fiquid theory. They should be both met~fic or insulating. In the next section, we an~yze these experimental resuRs from lhe fiewpoint of the ~pin~harge separation in the RVB state.
3. Theoret~al a n ~ y ~ s of the normal ~ate properties
Fi~t we briefly refiew the bafic ~ u r e s of RVB theory. The phyfic~ e ~ r o n oper~or C~. ~ expre~ed ~ the ~ave boson formalism as C~ =fi~b~,
(1)
where ~ (bi) are the annihilation operator of the fermion (boson) which represents the spinon (holon). There is local conservation of 'charge' which ~ expressed as ~'~ f ~ f ~ + b,*.bi = 1.
(2)
~r
The loc~ c o n f r o n t eq. (2) is closely related to the gauge invariance with respect to the following gauge ~ansformation: ~.---~ ~ e ~ ,
(3a)
bi --~ bi e ~ ,
(3b)
which leaves the phyfic~ e~ctron operator Ci. un-
810
~ Nagaosa / ~ w ~ e c ~ ~ R ~
changed. It is well known that the gauge fidd is ine~tably introduced to conserve the locM gauge invariance. This gauge field is explicit~ derived by confidering the Gausfian fluctuation around the RVB mean field solution. The phase of the RVB order parameters ( E ~ f ~ ) and (b~b~) constitute the space componen~ while the Lagrange mulfipl~r field A~~ the time component of the gauge field. Phys~M~, th~ gauge field represen~ the ~fing connecting the spinon and holon. It should be noted that this ~ring ~ confined in each of the two-dimensionM planes in contrast to the e ~ r o m a g n e t i c rid& The coupling of the gauge field with the spinon and holon ~ wri~en as
~ e o ~ in h ~ - ~
c~m~
the total system is the sum of those spinons and holons, i.e., (8)
P = Pv + P~ •
The re~stivity of the holon p~ dominates in eq. (8) except in the ove~doped region (x > Xm). Therefore, from eq. (7) the resistivity is proportional to the temperature T in the s~ange met~ region as ~ observed experimentMly. The H~I constant Ru ~ ~ven by R. -
R~X~ + R~Xv
,
(9)
XB + XF
L~., = ( j [ + j~)a~ ,
(4) and the thermopower S.b is ~ven by
where j ~ ( j ~ ) ~ the spinon (holon) current. As is evident ~om eq. (4), the integration over the gauge field gives the local c o n f r o n t
~ + i~ = o.
(5)
The bafic idea of the gauge field theory of highly correlated electron systems ~ to exchange the order of the integrations, that ~, to integrate the fermions and bosons first. This ~ves the dynamics of the gauge field, which ~ expressed by the propagator D ( q , ~ ) ~ven by D(q,o~)=(lo~l/q+x~q~+n~+n~)
-~ .
(6)
n~ (n~) ~ the superfluid density of the spinon (holon) sy~em while go ~ the sum of the dhmagnetic susceptib~ities of the spinon and holon, i.e., Xv + X~. In the case where both spinons and holons are in the normM phase, i.e., n~ = n~ = 0, the gauge field flu~uations have the appredable we~ht of the small frequency and long wavdength components. Th~ ~ves rise to the anomMous temperature dependence of various phyfical quantities. For example, the ~anspo~ fifetime rn of the boson scaHered by the gauge field fluctuations ~ evaluated as
1
T
z~
m~x ~
S,~ = S ~ + S ~ .
(10)
In summary, the p h y s ~ quantities within the plane can be obt~ned by c~culating the response of the spinons and holons in the presence of the flu~uating gauge field and combining them. On the other hand, the physic~ processes ~ong the c-a~s are e s s e n t i ~ different from those with~ the plane. The RVB state ~ r e ~ e d within each plane and the charge transfer between the planes occu~ only through the form of the physic~ d e ~ r o n . Therefore, the spinon and holon combine to form an d e ~ r o n , hop between the planes and ag~n di~odate into the spinon-holon pair. Hence, the phyfic~ prope~ies ~ong the c-a~s ~ve impo~ant information on the p h y s ~ electron of Green's function which ~ obt~ned by the convolution of those of the spinon and holon. As long as the hopping matrix e~ment ~ between the plane is sm~l enough and the temperature is not so low, we can treat ~ pe~urbative~, and the conducfiv~ ty ~c is ~ven by
o-,.=2e: ~::0~t~ ; 2~[-~Im
GR(k, E)]2
(7)
If either of the two systems shows the Meissner effect wRh superfluid den~ty Ov or p~, the small (q, w) fluctuations are suppressed compared with eq. (6). The response of the total system to the external electromagnetic field is obtained by taking into account the screening effect of the gauge field. Here we quote only the results. The readers are referred to the fiterature [5-7] for fu~her detai~. The re~stivity of
The phyfic~ electron spectr~ function has been obtained for the strange met~ state. It is composed of the quasi-particle peak and the broad background. If we put that resuR into eq. (11), we obt~n p,,~X/-~ reflecting the width of the quaff-particle peak. This exponent 1/2 ~ s m e a r than that for the Fermi liquid case where Pc ~ T~ Note that this metall~ temperature dependence occurs although the coherent motion along the c-axis ~ absent. By generali~ng the theoreti-
N. Nagaosa / New aspec~ of RVB theory in high-T~ cuprates c ~ k a m e w o r k d~cussed above to the spin gap and Fermi ~quid r e , o n , we examine the crossover phenomena in the n o r m ~ ~ r t e below. As has been discussed in section 1, the mean field phase ~anfition in fig. l(a) becomes the gradu~ crossover when the gauge field fluctuations are taken into account as shown in fig. l(b). However, the amplitude of the spinon p~fing (holon condensation) grows appre~ably in the low temperature r e ~ o n wilh low (high) x. In the "spin gap ~ a t e " , the pseudo gap appears in the denfity of ~ate of the spinon with the enhancement of lhe spinon diamagnet~ susceptibility Xv- This resul~ in the following consequences: (a) The resistivity p ~ within the plane d e b a t e s from the T-finear b e h a ~ o r to lower v~ues when the dimen~ o n ~ s s coupling constant g = 1/gam~ becomes ~ss than unity [7]. (b) The H ~ I constant R . approaches 1/x as the temperature ~ lowered. However, near the supercondu~ing T~, the flu~uation of the hoion condensation appears which resuRs in an increase in g~, and Rn a g ~ n decreases toward T~. Both X~ and gn dNerge at T~ and the H ~ I con~ant approaches the finite v~ue [x(1 + ~]-~ < x -~ with r = limr~X~/g~. Hence, the Hall con~ant has a peak above T~. Con~defing that the coherence ~ngth of the boson ~ short (of the order of the inte~holon di~ance) as well as the 2D nature of the system, it ~ reasonable that the fluctuation of the holon condensation is seen in the observable temperature r e , o n above T~. (c) The temperature dependence of the conductivity a,. ~ o n g the c-a~s becomes insulating because the phys~ electron s p e ~ r ~ fun~ion A(k, ~) has a (pseudo)gap when the p~fing of the spinon occurs. If the p~fing ~ ~wave with a gap A in the exotation spectrum p ~ = l / ~ . ~ e x p ( A / T ) . If the p~fing ~ dwave, on the other hand, the temperature dependence becomes weaker, i.e., p , . = l / a ~ l / T , because the depletion of the spectr~ we~ht ~ much less than the ~wave case. Th~ expl~ns the pos~ble correspondence between the (pseudo)gap in the magnet~ exotafions and the insulating temperature dependence of p,~ pointed out by Rice. These resuks are all in agreement with the experiments discu~ed in section 2. In p a ~ u ~ r , the drastic difference between p ~ and p~ is difficult to unde~tand ~ o m the viewpoint of Fermi fiquid theory. R can be n a t u r ~ expl~ned, on the other hand, assuming that the charge carriers within the plane are the holons whi~ those between the planes are the compo~te particles of the spinon and holon. The magnetic properties are m~nly determined by the spinons and will directly reflect the opening of the gap in t h o r exotation ~pectrum as discussed by Tanamoto et ~ . [4].
811
Now we an~yze the h ~ h concentration r e , o n , i.e. x > x m. In this case, the ampfitude of the holon condensation develops in the low temperature r e , o n and the phase flu~uations make the expe~ation v~ue ( b ) vanish. The correlation length ¢ in this case is determined by the inter-vortex di~ance wh~h grows e x p o n e n t i ~ as the temperature ~ lowered. In this r e , o n , the conductivity ga and the diamagnetic su~ ceptibifity g~ overwhelm those of the spinons, and the response of the t o t ~ sy~em is that of the spinon sy~em. When the corrdation ~ngth ¢ becomes longer than the t h e r m ~ ~ngth h v ~ T of the spinon, the holon can be regarded as bose condensed and the sy~em becomes indistinguishab~ from the Fermi liquid. In th~ case, the propagator of the gauge field ~ not fingular in the infrared fimR as d~cussed in eq. (6), and the resistifity of the spinon ~ proportion~ to T 2 as in the case of usu~ Fermi ~quid. Because the change from the ~range m e t ~ to the Fermi ~quid ~ a crossover, the resistivity p ~ changes g r a d u ~ from T-~near to T ~ behavior as the temperature ~ lowered a n d / o r x ~ increased. Therefore, we regard the temperature dependence P~b ~ T~ (a = 1.5) as the chara~ tefistic behafior of the intermediate r e , o n of the crossover from T to T~ The Hall constant is ~so determined by the spinon sy~em and ~ predicted to be negative and temperature independent. The conductivity g~ along the c-axis ~ven in eq. (11) ~ proportional to the ~fetime of the physic~ d e ~ r o n at the Fermi surface and ~ ~so proportional to T ~ These features are ~so in qualitat~e agreement wilh the e x p e r i m e n t . In condufion, we have d~cussed the crossover phenomena of ~ a n s p o ~ properties in the norm~ phase taking into account the fluctuations of spinon p~fing and holon condensation. The anom~ous features in the n o r m ~ phase can be n a t u r ~ unde~tood in terms cf spin-Charge separation in the RVB state. The supe~ conducting properties cf this s p i n ~ a r g e separated sy~em ~ le~ for fulure i n v e ~ a t i o n s .
Ackv owledgemen~ The author acknowledg~ E A . Lee, S. U c ~ d a , H. Fukuyama and T.M. Rice ~ r ~ u s ~ o n s .
Re~renc~ [1] D.M. Ginsburg, Phy~cM Properties of High Temper~ ture Supercondu~o~ H (World Scientific, Singapore, 1990); H. Fukuyama et ~., Strong CorreCtion and Superconducfiv~y (Spfinger-Verlag, Berlin, 1989).
812 [21 [3] [4] [5] [6]
N. Nagaosa / New aspec~ of RVB theory in high-T~, cuprates P.W. Anderson, Soence 235 (1987) 1196. Y. Suzumura et al., J. Phys. Soc. Jpn. 57 (1988) 2768. T. Tanamoto et ~., J. Phys. Soc. Jpn. 60 (1991) 3072. L. Ioffe and A. Larkin, Phys. Rev. B 39 (1989) 8988. N. Nagaosa and EA. Lee, Phys. Rev. Le~. 64 (1990) 2450.
[7] [8] [2 [10]
N. Nagaosa and EA. Lee, Phys. Rev. B 43 (1991) 1233. N. Nagaosa and EA. Lee, Phys. Rev. B 45 (1992) 966. S Uchida et ~., preprints 1992. T.M. Rice, private communications.
Phyfica B 186-188 (1993) 808-812 NortHland
Some new aspects of R V B theory in high-T c cuprates Naoto Nagaosa D ~ t m e m of App~d P h y ~ , Un~ersi~ of Tokyo, Hongo, B u n ~ k u , Tokyo 113, Japan We discuss some new aspec~ of RVB theory. By ta~ng into account the fluctuations of the spinon pMfing and the holon condensation, the crossover phenomena in the normM phase of high-T~ superconductors are clarified and compared with recent experiments.
I. Introduction
It is now well estabfished that the anom~ous properties of high-T, s u p e r c o n d u ~ o ~ are a~fibuted to strong electron correlations [1]. One of the most impo~ant t h e o r e t ~ issue is the posfibility of the breakdown of the Fermi hquid theory in this high~ correlated d e ~ r o n sy~em. The resonating v~ence bond (RVB) state first proposed by Anderson [2] shows non-Fermi liquid b e h a ~ o r where s ~ n and charge degrees of ~ e e d o m are somehow separated, and carried by two kinds of quaff particle, i.e. spinon and holon, respective~. The mean field theory for this spin~harge separated sy~em predicts an intere~ing phase diagram wh~h is schematically shown in fig. l(a) [3]. In r e , o n I, neRher the spinon p~fing nor the holon condensation occur. The n o r m ~ state properties of the "~range m e t ~ state" has been ~udied taking into account the gauge field fluctuations. In r e , o n II, only holon condensation occurs wh~e the spinon rem ~ n s in the n o r m ~ state. This state is a Fermi liquid fimHar to the heavy fermion. In r e , o n III, only the spinon p~fing occurs while holon rem~ns normS. This ~ cal~d the "spin gap state" because the gap in the spin exaltation is introduced due to p~fing fim~ar to the BCS state. Tanamoto et ~ . [4] have recent~ investigated the magnetic properties of the spinons in this spin gap state as well as the s~ange metal state and concluded that the NMR and neutron sc~tefing data can be expl~ned by the spinon with the dispe~ fion ~ v e n by the mean field theory if exchange enhancement ~ taken into account. However, spinons and holons are not independent of each other. They
Correspondence to: N. Nagaosa, Depa~ment of Applied Phy~cs, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan.
are c o u p ~ d through the local c o n f r o n t that the sum of the spinon number and holon number should be equal to 1 at every ~te. This c o n f r o n t ~ conveniently expressed in terms of the gauge field, and the meaning of the spin-charge separation has been investigated in
s~r~.gomota~ J
(~)
~
/b)
TD
~ ~ d
"\,
//
\'\,,/../" .~. ~
~
TBE
) g Fig. 1. (a)The mean-fidd phase diagram. (b) Modified phase diagram when in~astic fifetime effe~s are taken into account. The only true phase ~ansition ~ across the heavy line into the supercondu~ing phase.
0921-4526/93/$06~0 © 1993 - Ege~er Sdence Pubfishers B.V. All fights reserved
~ Na~a
/ New a ~
~ r m s of gauge fie~ theory ~ - ~ . In t~s language, the s~n~harge separation co~esponds to the deconfi~ng phase while the Fermi fiq~d co~esponds to the confin~g phase. At f i ~ ~ m p e r ~ u r e , lhe coupling with the gauge fie~ has been taken ~ t o account by the G~gzburg-Landau (GL) theory ~ d u d ~ g the two order p a r a m e ~ , i.e., those of the spinon p~ring and h ~ o n condensation [~. One condufion is t h ~ the mean field ~ansition of the s~non p~fing ( h o r n condensation) without h o r n condensation ~ n o n p~ring) becomes a crossover and the true phase tranfition is lhe supe~onducting one ~ wh~h both the order param~ers become nonzero fimuRaneou~y as shown in fig. l(b). ARhough there ~ no phase ~anfition in the norm~ phase, R is impo~ant to study the change ~ the physic~ properties across these crossovers because sever~ experiment~ resuRs show ~teresting beha~or as the concen~ation is changed. In t~s paper, we focus on the Uanspo~ properties in the n o r m ~ phase. The pre~ous an~ysis of the ~range metal r e , o n is extended to the s~n gap as well as the Fermi l ~ d r e , o n . The ~anspo~ properties ~ong the c-a~s are ~so ~scu~ed as well as the ~-plane properties.
2. Summary of some experimental results
Before discussing the theoretical results, we ~ummarize the relevant expefiment~ resuks recently obtained by Uchida et ~. [9]. They have measured: (1) the resistivity p~o in the plane; and (2) p~ ~ong the c-axis; (3) the H~I con~ant in the plane (with the magnetic field perpendicular to the plane); (4) the thermopower S,o parallel to the plane; (5) S~ ~ong the c-ax~ cf the LSCO tingle ~ r y ~ chan~ng the hole concentration. The resul~ are: (1) When the hole concentration x is s m e a r than the optim~ (highe~ Tc) concen~ation Xm=0.15 , ~ b begins to deviate from T-finear to the lower v~ue as the temperature is lowered below T = 150K (for x = 0.1). For x = Xm, the T-hnear beha¼or ~ observed down to the superconducting To. When x exceeds x m appredably, the refistivity has a pofitive curvature, i.e., Pab ~ T~ (a = 1.5). (2) The behavior of p~ changes dramatic~ly as the concentration x increases. In the low x region, the temperature dependence h insulating and the absolute value is much larger than the MoR fimit. As x ~ increased, the temperature dependence becomes weaker and weaker, and the absolute v~ue decreases and approaches the Mo~ hmit. In the overdoped region, the temperature dependence ~ met~hc and the absolute v~ue ~ less than the Mo~ hmR. Rice [10]
~ RVB ~
~ h ~ h - ~ cupmtes
~9
has pointed out the possible co~espondence of the insulating beha¼or of the re~stifity and the e ~ e n c e of the pseudo gap in the magnet~ exaltations. (3) The H~I constant decreases as the concentration is increased. It is positive for x < 0.3 and turns negative for x > 0.3. In the low concentration r e , o n , the absolute v~ue ~ large and has a peak above the superconducting T~. Near the optimal concentration x m, it ~ continuously decreasing with temperature. For x > 0.3, the Hall constant ~ negative and ~most temperature independent. (4) Thermopower paralld to the plane ~ positive with the absolute value ~ O . l k B / e . There ~ ~so a report that S.~ is much less sen~tNe to a large magnetic field than that expe~ed from the spin splitting effect. (5) Thermopower S~ ~ong the c-a~s ~ ~so positive and approximate~ expre~ed as S~ ~ ( k B / e ) ( T / 2 0 0 K) for x =0.15. These expefiment~ features seem to violate the Fermi fiquid pi~ure. P a ~ u l a d y Anderson d ~ m e d that the met~l~ behafior within the plane and the insulating behafior between the plane are not compatib~ with each other within the Fermi fiquid theory. They should be both met~fic or insulating. In the next section, we an~yze these experimental resuRs from lhe fiewpoint of the ~pin~harge separation in the RVB state.
3. Theoret~al a n ~ y ~ s of the normal ~ate properties
Fi~t we briefly refiew the bafic ~ u r e s of RVB theory. The phyfic~ e ~ r o n oper~or C~. ~ expre~ed ~ the ~ave boson formalism as C~ =fi~b~,
(1)
where ~ (bi) are the annihilation operator of the fermion (boson) which represents the spinon (holon). There is local conservation of 'charge' which ~ expressed as ~'~ f ~ f ~ + b,*.bi = 1.
(2)
~r
The loc~ c o n f r o n t eq. (2) is closely related to the gauge invariance with respect to the following gauge ~ansformation: ~.---~ ~ e ~ ,
(3a)
bi --~ bi e ~ ,
(3b)
which leaves the phyfic~ e~ctron operator Ci. un-
810
~ Nagaosa / ~ w ~ e c ~ ~ R ~
changed. It is well known that the gauge fidd is ine~tably introduced to conserve the locM gauge invariance. This gauge field is explicit~ derived by confidering the Gausfian fluctuation around the RVB mean field solution. The phase of the RVB order parameters ( E ~ f ~ ) and (b~b~) constitute the space componen~ while the Lagrange mulfipl~r field A~~ the time component of the gauge field. Phys~M~, th~ gauge field represen~ the ~fing connecting the spinon and holon. It should be noted that this ~ring ~ confined in each of the two-dimensionM planes in contrast to the e ~ r o m a g n e t i c rid& The coupling of the gauge field with the spinon and holon ~ wri~en as
~ e o ~ in h ~ - ~
c~m~
the total system is the sum of those spinons and holons, i.e., (8)
P = Pv + P~ •
The re~stivity of the holon p~ dominates in eq. (8) except in the ove~doped region (x > Xm). Therefore, from eq. (7) the resistivity is proportional to the temperature T in the s~ange met~ region as ~ observed experimentMly. The H~I constant Ru ~ ~ven by R. -
R~X~ + R~Xv
,
(9)
XB + XF
L~., = ( j [ + j~)a~ ,
(4) and the thermopower S.b is ~ven by
where j ~ ( j ~ ) ~ the spinon (holon) current. As is evident ~om eq. (4), the integration over the gauge field gives the local c o n f r o n t
~ + i~ = o.
(5)
The bafic idea of the gauge field theory of highly correlated electron systems ~ to exchange the order of the integrations, that ~, to integrate the fermions and bosons first. This ~ves the dynamics of the gauge field, which ~ expressed by the propagator D ( q , ~ ) ~ven by D(q,o~)=(lo~l/q+x~q~+n~+n~)
-~ .
(6)
n~ (n~) ~ the superfluid density of the spinon (holon) sy~em while go ~ the sum of the dhmagnetic susceptib~ities of the spinon and holon, i.e., Xv + X~. In the case where both spinons and holons are in the normM phase, i.e., n~ = n~ = 0, the gauge field flu~uations have the appredable we~ht of the small frequency and long wavdength components. Th~ ~ves rise to the anomMous temperature dependence of various phyfical quantities. For example, the ~anspo~ fifetime rn of the boson scaHered by the gauge field fluctuations ~ evaluated as
1
T
z~
m~x ~
S,~ = S ~ + S ~ .
(10)
In summary, the p h y s ~ quantities within the plane can be obt~ned by c~culating the response of the spinons and holons in the presence of the flu~uating gauge field and combining them. On the other hand, the physic~ processes ~ong the c-a~s are e s s e n t i ~ different from those with~ the plane. The RVB state ~ r e ~ e d within each plane and the charge transfer between the planes occu~ only through the form of the physic~ d e ~ r o n . Therefore, the spinon and holon combine to form an d e ~ r o n , hop between the planes and ag~n di~odate into the spinon-holon pair. Hence, the phyfic~ prope~ies ~ong the c-a~s ~ve impo~ant information on the p h y s ~ electron of Green's function which ~ obt~ned by the convolution of those of the spinon and holon. As long as the hopping matrix e~ment ~ between the plane is sm~l enough and the temperature is not so low, we can treat ~ pe~urbative~, and the conducfiv~ ty ~c is ~ven by
o-,.=2e: ~::0~t~ ; 2~[-~Im
GR(k, E)]2
(7)
If either of the two systems shows the Meissner effect wRh superfluid den~ty Ov or p~, the small (q, w) fluctuations are suppressed compared with eq. (6). The response of the total system to the external electromagnetic field is obtained by taking into account the screening effect of the gauge field. Here we quote only the results. The readers are referred to the fiterature [5-7] for fu~her detai~. The re~stivity of
The phyfic~ electron spectr~ function has been obtained for the strange met~ state. It is composed of the quasi-particle peak and the broad background. If we put that resuR into eq. (11), we obt~n p,,~X/-~ reflecting the width of the quaff-particle peak. This exponent 1/2 ~ s m e a r than that for the Fermi liquid case where Pc ~ T~ Note that this metall~ temperature dependence occurs although the coherent motion along the c-axis ~ absent. By generali~ng the theoreti-
N. Nagaosa / New aspec~ of RVB theory in high-T~ cuprates c ~ k a m e w o r k d~cussed above to the spin gap and Fermi ~quid r e , o n , we examine the crossover phenomena in the n o r m ~ ~ r t e below. As has been discussed in section 1, the mean field phase ~anfition in fig. l(a) becomes the gradu~ crossover when the gauge field fluctuations are taken into account as shown in fig. l(b). However, the amplitude of the spinon p~fing (holon condensation) grows appre~ably in the low temperature r e ~ o n wilh low (high) x. In the "spin gap ~ a t e " , the pseudo gap appears in the denfity of ~ate of the spinon with the enhancement of lhe spinon diamagnet~ susceptibility Xv- This resul~ in the following consequences: (a) The resistivity p ~ within the plane d e b a t e s from the T-finear b e h a ~ o r to lower v~ues when the dimen~ o n ~ s s coupling constant g = 1/gam~ becomes ~ss than unity [7]. (b) The H ~ I constant R . approaches 1/x as the temperature ~ lowered. However, near the supercondu~ing T~, the flu~uation of the hoion condensation appears which resuRs in an increase in g~, and Rn a g ~ n decreases toward T~. Both X~ and gn dNerge at T~ and the H ~ I con~ant approaches the finite v~ue [x(1 + ~]-~ < x -~ with r = limr~X~/g~. Hence, the Hall con~ant has a peak above T~. Con~defing that the coherence ~ngth of the boson ~ short (of the order of the inte~holon di~ance) as well as the 2D nature of the system, it ~ reasonable that the fluctuation of the holon condensation is seen in the observable temperature r e , o n above T~. (c) The temperature dependence of the conductivity a,. ~ o n g the c-a~s becomes insulating because the phys~ electron s p e ~ r ~ fun~ion A(k, ~) has a (pseudo)gap when the p~fing of the spinon occurs. If the p~fing ~ ~wave with a gap A in the exotation spectrum p ~ = l / ~ . ~ e x p ( A / T ) . If the p~fing ~ dwave, on the other hand, the temperature dependence becomes weaker, i.e., p , . = l / a ~ l / T , because the depletion of the spectr~ we~ht ~ much less than the ~wave case. Th~ expl~ns the pos~ble correspondence between the (pseudo)gap in the magnet~ exotafions and the insulating temperature dependence of p,~ pointed out by Rice. These resuks are all in agreement with the experiments discu~ed in section 2. In p a ~ u ~ r , the drastic difference between p ~ and p~ is difficult to unde~tand ~ o m the viewpoint of Fermi fiquid theory. R can be n a t u r ~ expl~ned, on the other hand, assuming that the charge carriers within the plane are the holons whi~ those between the planes are the compo~te particles of the spinon and holon. The magnetic properties are m~nly determined by the spinons and will directly reflect the opening of the gap in t h o r exotation ~pectrum as discussed by Tanamoto et ~ . [4].
811
Now we an~yze the h ~ h concentration r e , o n , i.e. x > x m. In this case, the ampfitude of the holon condensation develops in the low temperature r e , o n and the phase flu~uations make the expe~ation v~ue ( b ) vanish. The correlation length ¢ in this case is determined by the inter-vortex di~ance wh~h grows e x p o n e n t i ~ as the temperature ~ lowered. In this r e , o n , the conductivity ga and the diamagnetic su~ ceptibifity g~ overwhelm those of the spinons, and the response of the t o t ~ sy~em is that of the spinon sy~em. When the corrdation ~ngth ¢ becomes longer than the t h e r m ~ ~ngth h v ~ T of the spinon, the holon can be regarded as bose condensed and the sy~em becomes indistinguishab~ from the Fermi liquid. In th~ case, the propagator of the gauge field ~ not fingular in the infrared fimR as d~cussed in eq. (6), and the resistifity of the spinon ~ proportion~ to T 2 as in the case of usu~ Fermi ~quid. Because the change from the ~range m e t ~ to the Fermi ~quid ~ a crossover, the resistivity p ~ changes g r a d u ~ from T-~near to T ~ behavior as the temperature ~ lowered a n d / o r x ~ increased. Therefore, we regard the temperature dependence P~b ~ T~ (a = 1.5) as the chara~ tefistic behafior of the intermediate r e , o n of the crossover from T to T~ The Hall constant is ~so determined by the spinon sy~em and ~ predicted to be negative and temperature independent. The conductivity g~ along the c-axis ~ven in eq. (11) ~ proportional to the ~fetime of the physic~ d e ~ r o n at the Fermi surface and ~ ~so proportional to T ~ These features are ~so in qualitat~e agreement wilh the e x p e r i m e n t . In condufion, we have d~cussed the crossover phenomena of ~ a n s p o ~ properties in the norm~ phase taking into account the fluctuations of spinon p~fing and holon condensation. The anom~ous features in the n o r m ~ phase can be n a t u r ~ unde~tood in terms cf spin-Charge separation in the RVB state. The supe~ conducting properties cf this s p i n ~ a r g e separated sy~em ~ le~ for fulure i n v e ~ a t i o n s .
Ackv owledgemen~ The author acknowledg~ E A . Lee, S. U c ~ d a , H. Fukuyama and T.M. Rice ~ r ~ u s ~ o n s .
Re~renc~ [1] D.M. Ginsburg, Phy~cM Properties of High Temper~ ture Supercondu~o~ H (World Scientific, Singapore, 1990); H. Fukuyama et ~., Strong CorreCtion and Superconducfiv~y (Spfinger-Verlag, Berlin, 1989).
812 [21 [3] [4] [5] [6]
N. Nagaosa / New aspec~ of RVB theory in high-T~, cuprates P.W. Anderson, Soence 235 (1987) 1196. Y. Suzumura et al., J. Phys. Soc. Jpn. 57 (1988) 2768. T. Tanamoto et ~., J. Phys. Soc. Jpn. 60 (1991) 3072. L. Ioffe and A. Larkin, Phys. Rev. B 39 (1989) 8988. N. Nagaosa and EA. Lee, Phys. Rev. Le~. 64 (1990) 2450.
[7] [8] [2 [10]
N. Nagaosa and EA. Lee, Phys. Rev. B 43 (1991) 1233. N. Nagaosa and EA. Lee, Phys. Rev. B 45 (1992) 966. S Uchida et ~., preprints 1992. T.M. Rice, private communications.