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Excitation spectra and bose condensation in superconductors with strong correlation
Physica 148B (1987) 391-395 North-Holland, Amsterdam
E X C I T A T I O N S P E C T R A AND BOSE C O N D E N S A T I O N IN S U P E R C O N D U C T O R S W I T H STRONG CORRELATION Y. ISAWA, S. M A E K A W A * and H. EBISAWA** Research Institute of Electrical Communication, Tohoku University, Sendai 980, Japan * Institute for Materials Research, Tohoku University, Sendal 980, Japan ** Department of Engineering Science, Tohoku University, Sendal 980, Japan
Received 7 August 1987 We study the pairing mechanism of high T~ oxides based on the Hubbard model. The transformation of the electron operators to a set of four operators is introduced for the analysis. We discuss the condensation of the resonating valence bond and the Bose condensation responsible for the superconductivity together with the excitation spectra. The effects of the long range intersite repulsive interaction which causes a dramatic change in the hole excitation spectra is also discussed. The formulas for the specific heat at low temperatures and the resistivity in normal state are given. I. Introduction The discovery of a n u m b e r of new high T~ oxides has stimulated the study of unconventional pairing mechanisms. H e r e we investigate a pairing mechanism coming from the strong e l e c t r o n - e l e c t r o n correlation by use of the H u b b a r d model. We introduce a new o p e r a t o r transformation: Electron operators at each site are represented by a set of four atomic operators. T h e y are related to four atomic eigenstates: (i) e m p t y state with no electron, (ii) doubly occupied state with two electrons, and (iii) two singly occupied states with spins up and down each of which has one electron. F r o m the theoretical analysis of H u b b a r d Hamiltonian rewritten by use of atomic operators, we find that two types of order p a r a m e t e r s appear. O n e of them results from Bose condensation of the charged bosons (holes) in e m p t y states, while the other from the singly occupied states. The latter is identified with the condensation in the resonating valence bond ( R V B ) state p r o p o s e d by A n d e r s o n [1-3]. Since the order p a r a m e t e r responsible for the superconductivity is a product of these two types of atomic order p a r a m e t e r s in real space and time representation, the critical t e m p e r a t u r e of superconductivity, To, is equal to the lower critical t e m p e r a t u r e of two atomic orderings. The RVB condensation occurs even in the absense of holes, while the hole condensation does not. Hence T c is equal to Tc(h ) (To(h): the critical t e m p e r a t u r e of hole condensation) when the density of holes is dilute. We find that our o p e r a t o r transformation appropriately describes the properties of holes and spinons proposed by A n d e r s o n et al. [1-3]. 2. Hubbard model and operator transformation The H u b b a r d Hamiltonian is given by +
n = 2 ~iCeriC~i+ cri
q- 2 t i j c ~ i c ~ j - [ - 1 U 2 n ~ i n - ~ i (i ,j )cr icr
,
(1)
where i labels the atomic site, o- is spin, c ~ is the annihilation o p e r a t o r for the electron, %i is the site energy measured from the chemical p o t e n t i a l / z , t~j is the transfer matrix and U is the on-site C o u l o m b repulsion. 0378-4363/87/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation
Y. Isawa et al. / Superconductors with strong correlation
392
We introduce the operator transformation in which the fermion field operators are expressed by a combination of a set of four a t o m i c operators, + i -I- o-a_o_ia2i + Co.i ~--- aoiao,
,
+i ---- ao_iaoi + + Co. -4- ora2ia
(o-= -+1),
~,i ,
(2)
where aoi, a~i and a2i are the annihilation operators for the empty state, the singly occupied state with spin ~r, and the doubly occupied state in the /-site, respectively. The operators a ~ are fermion operators, and a0~ and azi are boson operators. The local constraint in each site is given by + + + aoiaoi + E~ a~ia~i + a2~azi = 1. This operator transformation was proposed by one of the authors in 1979 [4] for the analysis of the H u b b a r d and the Anderson models. Similar transformations called slave boson methods have been proposed [5-8]. In terms of this atomic operator transformation, the electron + + + number n~ at /-site is given by n~ = Z~ c ~ c ~ = E~ a~ia~ + 2aziazi. How can we assign the spin and the charge of the electron to the atomic operators? The operator a~ is the one with spin ½ since the spin a ~+a ~ iandSzi = ~(a.~ia,~ components are given b y S + = a ~+a ~ , S i - = 1 + _ a ~+a ~ ) . H e r e w e w r i t e 1' for cr = 1 and $ for o- = - 1 . We take the system to be neutral when the band is half-filled. Then the total charge operator Q is defined by Q = Zi Q~ = E~ (a2+ia2i- ao]ao~ ) with the local charge Q~. Thus the singly occupied state has no charge, i.e. it is neutral, the empty state has a charge opposite to the electron and the doubly occupied state has a charge equal to the electron. In this sense, an empty site will be equivalent to a charged boson (hole) at the site introduced by Anderson et al. [2]. Though another charged boson related to the doubly occupied state is not important in the system with strong on-site Coulomb interaction, it will play an essential role when U is comparable to the transfer matrix t. The order parameter responsible for the singlet superconductivity on discrete lattice is given by
(c,~ic_,~j) = { ao+~a,~ao+ja ,~j ) - or ( ao+ia,~a,~+ja2j) + o" ( a +_,~ia2~ao+ia_,~,) - ( a s,~a2ia,~+ja2j ) .
(3)
The first term in r.h.s, of (3) is decoupled to be a product of two terms. One is the anomalous function relevant to empty state (hole). The other relevant to the singly occupied states, which we identify with spinon [1-3], is the order parameter of the RVB condensation. The remaining terms of (3) are negligible when U >> t. In the frequency and m o m e n t u m representation, (3) is given by the convolution of two anomalous correlation functions, which is in a remarkable constrast with the conventional BCS theory. The Hubbard Hamiltonian is written as follows [4]: +
+
+
H = ~, ( G i - A)a+ia,,i + ~'~ (e2i - A)aeia2i - A ~'~ aoiaoi + Z i,cr
i +
+
i +
+
-F ora o,ia_o_jaoia2j -F o-a2iaoja_o_ia o.j) ,
+
+
+
tq(a~iaojaoia~j - a2ia-~ja-~ia2j
i¢-j,~r
(4)
where 62i /~o-i-}- e--~i q- U. H e r e the local constraint is taken into account on the average through a Lagrange multiplier A on the assumption that it is equal in each site. The excitation spectrum of the spinon is given by E 2 ( p ) = e(p) 2 + 41t°2(p)] 2 with the spinon energy in the absence of RVB condensation, e ( p ) = e - A + t°°(p) - tZZ(p). H e r e t°°(p), t22(p), and t°2(p) are the Fourier transform of tij(aoiao+j), %i(az+ja2i) and (%~(aojazg)+ t~(ao~azj))/2, respectively, e ( p ) is small when (i) both the empty and the doubly occupied sites are few, or (ii) the populations of the empty and the doubly occupied states are equal. In the half-filled case, e(p) vanishes since there are no holes and no doubly occupied sites so that t ° ° ( p ) = t 2 2 ( p ) = 0 and e = A. The spinon correlation functions are given by =
Y. Isawa et al. / Superconductors +
+
o'(%pa~_p) -
+
E(p) th(/3E(p)/2),
[
(a,~p%e) : ½ 1
s(p) th(/3E(p)/2) E(p)
with strong correlation
(t2°(p) : t°2(p)*),
]
393
(5) (6)
with the inverse temperature/3. When U is much larger than the hole band width given by the Fourier transform of tij(a~i%j), + we obtain
(ae'a°J+ + ) -
U1 E,~ crt,i(aS, a+~i)(ao,ao+j) .
(7)
+
In the half-filled band, (a0~a0j) is replaced by 6zj. From (7), it is easy to see that the expectation value of H is, within the subspace of singly occupied states, locally Gauge invariant for the half-filled band whereas it is only globally Gauge invariant in the other cases [3]. If we consider the Gauge degrees of freedom for the whole set of 4 states located on each site, more extended discussion will be required. Let us take the square lattice and the transfer matrix t being nonzero only for the nearest neighbor hopping. Then we obtain t2°(P) = - ---U-]t[2[(1 + (b~b°t) + 41~012)(cos(pxd~) + cos(pydy))(R x + Ry) ,
+ (1 + (bo~bo,) )(cos(p~d x) - cos(pydy))(R x - Ry)*],
(8)
where a 0 is the condensed component of the hole (empty state), b 0 denotes the elementary excitation of the hole, dx,y and Rx,y are the lattice distances and the values of (R~j) = ( 1 / V ~ ) E~ o'(a~ia,j) between the nearest neighbor sites in the x and y directions, respectively. We discuss a 0 and b 0 in the next section. The self-consistent equation for (R x +- Ry) is given by
(Rx +- Ry) - - V ~
~ (coS(pxdx)+-coS(pydy))~t"2(p) - ~ th(/3E(p)/2).
(9)
3. Bose condensation The anomalous correlation functions (aoao) and ( a o a 2 ) resulting from the Bose condensation are responsible for the superconductivity. The boson operators in m o m e n t u m representation are written as aok = aO~Ok+ bok and a 2 k = ot2~0k+ bzk, where c-numbers a0* and a 2 are linearly dependent when on-site Coulomb repulsion is strong: a 2 = - E ~i o'Q (a_~ia~j) a ~ / U. The number of hole excitation is given by the Bose distribution with the excitation energy Eo(k ) = tll(k) - A - ]t-11(k)12/U. When a 0 ~ 0, A is fixed: A = t ' ( k = 0 ) - ]t-l~(k = 0)]2/U. Thus an elementary excitation mode Eo(k ) for hole and doubly occupied state has an energy tending to zero for long wavelength. This is well-known as the Goldstone theorem for the system with broken symmetry. The self-consistent equation for a 0 is given by
~, (a~+pa~p) + E (bo+kbok) + lao[ 2= Ns, per
k
(lo)
where N s is the total number of lattice sites. Here the population in doubly occupied state is ignored.
Y. lsawa et al. / Superconductors with strong correlation
394
From (10), the critical temperature of the Bose condensation, Tc(h), is given in 3-dimensional system (3-D) [2] by Tc(h ) _
2"rrh2 ( N s - - N c ~
2/3
m * k B \ 2.612/2 /
(11)
'
where N c and ~2 are the total number of spinon excitations for a 0 = 0 and the sample volume respectively, and k B is the Boltzmann constant. For the square lattice with nearest neighbor hopping, the hole mass at k ~ 0 , m*, is given by m* 1 =2td2(E~ (a+a,,) ..... + 16tR2/U) where E, (a+a~,) ..... is the correlation function of spinons located on the nearest neighbor sites and t is the transfer matrix.
4. Effects of intersite repulsive interaction By making use of the arbitrariness of the relative phases (or Gauge degrees of freedom) among atomic operators, the intersite repulsive interaction Vq is written as follows: Vi. t = ~ VqQiQ j .
(12)
(ij)
Since Vint is an interaction among charges, it does not directly have an effect on the spinon. Thus we find that the intersite Coulomb repulsion deteriorates neither the RVB condensation nor the Bose condensation as long as Vint < U. Hence the superconductivity will survive even if the attractive interaction are not introduced to cancel the intersite Coulomb repulsion. This is in contrast with the arguments by Hirsch [8], Ruckenstein et al. [6] and Inui et al. [7]. However, the hole excitation spectrum is strongly modified by V~nt. Since the doubly occupied site is few, Vint is the interaction among charged holes located on different sites. By applying the Bogoliubov approximation to the interacting charged boson system, we obain the hole excitation spectrum: W.O2h = 2 [ a o [ 2 V ( k ) g o ( k )
+
E0(k)2
(13)
where V(k) is the Fourier component of Vq. When V(k--> 0) is finite, the excitation mode is phonon like since ~ok ~ k at small k. If the intersite repulsive interaction in 3-D is of long range and is proportional to 1/r, i.e. V(k) oc N , / k 2, there appears a finite energy gap at k--> 0 corresponding to the plasma oscillation, and a local minimum at finite k. Thus the intersite repulsive interaction gives rise to a marked change of the hole excitation spectrum. However Tc(h ) is not affected, in the mean field approximation, by the interaction since it works only when a0 ~ 0. The excitation of holes is described by the following correlation functions: + (a,,kaok) = ½[ - 1 + ~°°(k) coth(/3~0(k)/2)] + la01260k,
(a,~
kao~) =
v(k) 2w(k) ~,2
where O~o(k) = Eo(k ) +
coth(3w(k)/2)
1012V(k).
+ ~2
a(lk '
(14)
(15)
Y. Isawa et al. / Superconductors with strong correlation
395
5. Specific heat at low temperatures and resistivity in normal state Let us assume that T c = Tc(h) and Tc(R ) > Tc(h ), where T~(R) is the critical temperature of the condensation in RVB state, For T < T c, the specific heat C v is given by 2
-rr
T Cv=TN,~=-~/T,
(16)
where T 0 = 12"trt2[R[(1 + c l a o ] Z / N s ) / V ~ k , U with c being a constant of O(1). The linear temperature dependence of C v is attributed to the constant spinon density of states at low energy region [2] which is justified at low hole density. Since both R and a 0 decrease as the temperature rises, C v / T is an increasing function of T. 7 will be a few mJ tool -~ K 2 for J R [ - 1 when tZ/u ~ 0.05(eV) and increases with increasing applied magnetic field H since ]R[ decreases with increasing H. Let us discuss the resistivity in normal state for Tc(h ) < T < Tc(R ). Hole has charge and moves almost freely. Thus hole carries current. Its lifetime is determined by elastic scattering due to spinons through the hopping. From the Drude like formula for the hole, the resistivity is given by
p=
m*(U/t)ZkB T 47rZn,e2R2h ,
(17)
where the hole density n* is low. If m* is equal to the free electron mass, n* = 102~(1/cm3), and t / U = 0.1, we obtain p = 130/R 2 (ix12 cm) at 100 K. A n o t h e r contribution to the conductivity, associated with, the scattering of the RVB condensed state due to hopping, will be essential as the temperature increases. We consider that the experimental results of the specific heat, C v ~ y T , at low temperature ( T ~ To) and the resistivity, p ~ T, in normal state imply the relation, T c = Tc(h ) and Tc(R ) > Tc(h), as discussed by Anderson et al. [2].
6. Summary The operator transformation developed in this paper is suitable for treating the dynamics of the charged boson (hole) and the spinon, introduced by Anderson, which give an appropriate description for the properties in highly correlated systems. We explicitly showed that the order parameter responsible for superconductivity results when the Bose condensation of the holes and the RVB condensation of pseudofermions (or spinons) occur at the same time. We find that the intersite repulsive interaction gives rise to a marked change of the hole excitation spectrum, which will modify the electron density of states. The long range part of the Coulomb interaction in addion to the short range part will be essential to the understanding of the various properties of high T c oxides. By applying our method to the specific heat at low temperatures and the resistivity in normal state, we obtain the results in agreement with the experimental ones.
References [1] [2] [3] [4] [5] [6] [7] [8]
EW. Anderson, Science 235 (1987) 1196. P.W. Anderson, G. Baskaran, Z. Zou and T. Hsu, Phys. Rev. Lett. 58 (1987) 2790. G. Baskaran and P.W. Anderson, preprint. Y. Isawa, Meeting at the Physical Society of Japan, 1979, unpublished. D.M. Newns, preprint. A.E. Ruckenstein, P.J. Hirschfeld and J. Appel, preprint M. Inui, S. Doniach, EJ. Hirschfeld and A.E. Ruckenstein, preprint. J.E. Hirsch, Phys. Rev. B 35 (1987) 8726.
E X C I T A T I O N S P E C T R A AND BOSE C O N D E N S A T I O N IN S U P E R C O N D U C T O R S W I T H STRONG CORRELATION Y. ISAWA, S. M A E K A W A * and H. EBISAWA** Research Institute of Electrical Communication, Tohoku University, Sendai 980, Japan * Institute for Materials Research, Tohoku University, Sendal 980, Japan ** Department of Engineering Science, Tohoku University, Sendal 980, Japan
Received 7 August 1987 We study the pairing mechanism of high T~ oxides based on the Hubbard model. The transformation of the electron operators to a set of four operators is introduced for the analysis. We discuss the condensation of the resonating valence bond and the Bose condensation responsible for the superconductivity together with the excitation spectra. The effects of the long range intersite repulsive interaction which causes a dramatic change in the hole excitation spectra is also discussed. The formulas for the specific heat at low temperatures and the resistivity in normal state are given. I. Introduction The discovery of a n u m b e r of new high T~ oxides has stimulated the study of unconventional pairing mechanisms. H e r e we investigate a pairing mechanism coming from the strong e l e c t r o n - e l e c t r o n correlation by use of the H u b b a r d model. We introduce a new o p e r a t o r transformation: Electron operators at each site are represented by a set of four atomic operators. T h e y are related to four atomic eigenstates: (i) e m p t y state with no electron, (ii) doubly occupied state with two electrons, and (iii) two singly occupied states with spins up and down each of which has one electron. F r o m the theoretical analysis of H u b b a r d Hamiltonian rewritten by use of atomic operators, we find that two types of order p a r a m e t e r s appear. O n e of them results from Bose condensation of the charged bosons (holes) in e m p t y states, while the other from the singly occupied states. The latter is identified with the condensation in the resonating valence bond ( R V B ) state p r o p o s e d by A n d e r s o n [1-3]. Since the order p a r a m e t e r responsible for the superconductivity is a product of these two types of atomic order p a r a m e t e r s in real space and time representation, the critical t e m p e r a t u r e of superconductivity, To, is equal to the lower critical t e m p e r a t u r e of two atomic orderings. The RVB condensation occurs even in the absense of holes, while the hole condensation does not. Hence T c is equal to Tc(h ) (To(h): the critical t e m p e r a t u r e of hole condensation) when the density of holes is dilute. We find that our o p e r a t o r transformation appropriately describes the properties of holes and spinons proposed by A n d e r s o n et al. [1-3]. 2. Hubbard model and operator transformation The H u b b a r d Hamiltonian is given by +
n = 2 ~iCeriC~i+ cri
q- 2 t i j c ~ i c ~ j - [ - 1 U 2 n ~ i n - ~ i (i ,j )cr icr
,
(1)
where i labels the atomic site, o- is spin, c ~ is the annihilation o p e r a t o r for the electron, %i is the site energy measured from the chemical p o t e n t i a l / z , t~j is the transfer matrix and U is the on-site C o u l o m b repulsion. 0378-4363/87/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation
Y. Isawa et al. / Superconductors with strong correlation
392
We introduce the operator transformation in which the fermion field operators are expressed by a combination of a set of four a t o m i c operators, + i -I- o-a_o_ia2i + Co.i ~--- aoiao,
,
+i ---- ao_iaoi + + Co. -4- ora2ia
(o-= -+1),
~,i ,
(2)
where aoi, a~i and a2i are the annihilation operators for the empty state, the singly occupied state with spin ~r, and the doubly occupied state in the /-site, respectively. The operators a ~ are fermion operators, and a0~ and azi are boson operators. The local constraint in each site is given by + + + aoiaoi + E~ a~ia~i + a2~azi = 1. This operator transformation was proposed by one of the authors in 1979 [4] for the analysis of the H u b b a r d and the Anderson models. Similar transformations called slave boson methods have been proposed [5-8]. In terms of this atomic operator transformation, the electron + + + number n~ at /-site is given by n~ = Z~ c ~ c ~ = E~ a~ia~ + 2aziazi. How can we assign the spin and the charge of the electron to the atomic operators? The operator a~ is the one with spin ½ since the spin a ~+a ~ iandSzi = ~(a.~ia,~ components are given b y S + = a ~+a ~ , S i - = 1 + _ a ~+a ~ ) . H e r e w e w r i t e 1' for cr = 1 and $ for o- = - 1 . We take the system to be neutral when the band is half-filled. Then the total charge operator Q is defined by Q = Zi Q~ = E~ (a2+ia2i- ao]ao~ ) with the local charge Q~. Thus the singly occupied state has no charge, i.e. it is neutral, the empty state has a charge opposite to the electron and the doubly occupied state has a charge equal to the electron. In this sense, an empty site will be equivalent to a charged boson (hole) at the site introduced by Anderson et al. [2]. Though another charged boson related to the doubly occupied state is not important in the system with strong on-site Coulomb interaction, it will play an essential role when U is comparable to the transfer matrix t. The order parameter responsible for the singlet superconductivity on discrete lattice is given by
(c,~ic_,~j) = { ao+~a,~ao+ja ,~j ) - or ( ao+ia,~a,~+ja2j) + o" ( a +_,~ia2~ao+ia_,~,) - ( a s,~a2ia,~+ja2j ) .
(3)
The first term in r.h.s, of (3) is decoupled to be a product of two terms. One is the anomalous function relevant to empty state (hole). The other relevant to the singly occupied states, which we identify with spinon [1-3], is the order parameter of the RVB condensation. The remaining terms of (3) are negligible when U >> t. In the frequency and m o m e n t u m representation, (3) is given by the convolution of two anomalous correlation functions, which is in a remarkable constrast with the conventional BCS theory. The Hubbard Hamiltonian is written as follows [4]: +
+
+
H = ~, ( G i - A)a+ia,,i + ~'~ (e2i - A)aeia2i - A ~'~ aoiaoi + Z i,cr
i +
+
i +
+
-F ora o,ia_o_jaoia2j -F o-a2iaoja_o_ia o.j) ,
+
+
+
tq(a~iaojaoia~j - a2ia-~ja-~ia2j
i¢-j,~r
(4)
where 62i /~o-i-}- e--~i q- U. H e r e the local constraint is taken into account on the average through a Lagrange multiplier A on the assumption that it is equal in each site. The excitation spectrum of the spinon is given by E 2 ( p ) = e(p) 2 + 41t°2(p)] 2 with the spinon energy in the absence of RVB condensation, e ( p ) = e - A + t°°(p) - tZZ(p). H e r e t°°(p), t22(p), and t°2(p) are the Fourier transform of tij(aoiao+j), %i(az+ja2i) and (%~(aojazg)+ t~(ao~azj))/2, respectively, e ( p ) is small when (i) both the empty and the doubly occupied sites are few, or (ii) the populations of the empty and the doubly occupied states are equal. In the half-filled case, e(p) vanishes since there are no holes and no doubly occupied sites so that t ° ° ( p ) = t 2 2 ( p ) = 0 and e = A. The spinon correlation functions are given by =
Y. Isawa et al. / Superconductors +
+
o'(%pa~_p) -
+
E(p) th(/3E(p)/2),
[
(a,~p%e) : ½ 1
s(p) th(/3E(p)/2) E(p)
with strong correlation
(t2°(p) : t°2(p)*),
]
393
(5) (6)
with the inverse temperature/3. When U is much larger than the hole band width given by the Fourier transform of tij(a~i%j), + we obtain
(ae'a°J+ + ) -
U1 E,~ crt,i(aS, a+~i)(ao,ao+j) .
(7)
+
In the half-filled band, (a0~a0j) is replaced by 6zj. From (7), it is easy to see that the expectation value of H is, within the subspace of singly occupied states, locally Gauge invariant for the half-filled band whereas it is only globally Gauge invariant in the other cases [3]. If we consider the Gauge degrees of freedom for the whole set of 4 states located on each site, more extended discussion will be required. Let us take the square lattice and the transfer matrix t being nonzero only for the nearest neighbor hopping. Then we obtain t2°(P) = - ---U-]t[2[(1 + (b~b°t) + 41~012)(cos(pxd~) + cos(pydy))(R x + Ry) ,
+ (1 + (bo~bo,) )(cos(p~d x) - cos(pydy))(R x - Ry)*],
(8)
where a 0 is the condensed component of the hole (empty state), b 0 denotes the elementary excitation of the hole, dx,y and Rx,y are the lattice distances and the values of (R~j) = ( 1 / V ~ ) E~ o'(a~ia,j) between the nearest neighbor sites in the x and y directions, respectively. We discuss a 0 and b 0 in the next section. The self-consistent equation for (R x +- Ry) is given by
(Rx +- Ry) - - V ~
~ (coS(pxdx)+-coS(pydy))~t"2(p) - ~ th(/3E(p)/2).
(9)
3. Bose condensation The anomalous correlation functions (aoao) and ( a o a 2 ) resulting from the Bose condensation are responsible for the superconductivity. The boson operators in m o m e n t u m representation are written as aok = aO~Ok+ bok and a 2 k = ot2~0k+ bzk, where c-numbers a0* and a 2 are linearly dependent when on-site Coulomb repulsion is strong: a 2 = - E ~i o'Q (a_~ia~j) a ~ / U. The number of hole excitation is given by the Bose distribution with the excitation energy Eo(k ) = tll(k) - A - ]t-11(k)12/U. When a 0 ~ 0, A is fixed: A = t ' ( k = 0 ) - ]t-l~(k = 0)]2/U. Thus an elementary excitation mode Eo(k ) for hole and doubly occupied state has an energy tending to zero for long wavelength. This is well-known as the Goldstone theorem for the system with broken symmetry. The self-consistent equation for a 0 is given by
~, (a~+pa~p) + E (bo+kbok) + lao[ 2= Ns, per
k
(lo)
where N s is the total number of lattice sites. Here the population in doubly occupied state is ignored.
Y. lsawa et al. / Superconductors with strong correlation
394
From (10), the critical temperature of the Bose condensation, Tc(h), is given in 3-dimensional system (3-D) [2] by Tc(h ) _
2"rrh2 ( N s - - N c ~
2/3
m * k B \ 2.612/2 /
(11)
'
where N c and ~2 are the total number of spinon excitations for a 0 = 0 and the sample volume respectively, and k B is the Boltzmann constant. For the square lattice with nearest neighbor hopping, the hole mass at k ~ 0 , m*, is given by m* 1 =2td2(E~ (a+a,,) ..... + 16tR2/U) where E, (a+a~,) ..... is the correlation function of spinons located on the nearest neighbor sites and t is the transfer matrix.
4. Effects of intersite repulsive interaction By making use of the arbitrariness of the relative phases (or Gauge degrees of freedom) among atomic operators, the intersite repulsive interaction Vq is written as follows: Vi. t = ~ VqQiQ j .
(12)
(ij)
Since Vint is an interaction among charges, it does not directly have an effect on the spinon. Thus we find that the intersite Coulomb repulsion deteriorates neither the RVB condensation nor the Bose condensation as long as Vint < U. Hence the superconductivity will survive even if the attractive interaction are not introduced to cancel the intersite Coulomb repulsion. This is in contrast with the arguments by Hirsch [8], Ruckenstein et al. [6] and Inui et al. [7]. However, the hole excitation spectrum is strongly modified by V~nt. Since the doubly occupied site is few, Vint is the interaction among charged holes located on different sites. By applying the Bogoliubov approximation to the interacting charged boson system, we obain the hole excitation spectrum: W.O2h = 2 [ a o [ 2 V ( k ) g o ( k )
+
E0(k)2
(13)
where V(k) is the Fourier component of Vq. When V(k--> 0) is finite, the excitation mode is phonon like since ~ok ~ k at small k. If the intersite repulsive interaction in 3-D is of long range and is proportional to 1/r, i.e. V(k) oc N , / k 2, there appears a finite energy gap at k--> 0 corresponding to the plasma oscillation, and a local minimum at finite k. Thus the intersite repulsive interaction gives rise to a marked change of the hole excitation spectrum. However Tc(h ) is not affected, in the mean field approximation, by the interaction since it works only when a0 ~ 0. The excitation of holes is described by the following correlation functions: + (a,,kaok) = ½[ - 1 + ~°°(k) coth(/3~0(k)/2)] + la01260k,
(a,~
kao~) =
v(k) 2w(k) ~,2
where O~o(k) = Eo(k ) +
coth(3w(k)/2)
1012V(k).
+ ~2
a(lk '
(14)
(15)
Y. Isawa et al. / Superconductors with strong correlation
395
5. Specific heat at low temperatures and resistivity in normal state Let us assume that T c = Tc(h) and Tc(R ) > Tc(h ), where T~(R) is the critical temperature of the condensation in RVB state, For T < T c, the specific heat C v is given by 2
-rr
T Cv=TN,~=-~/T,
(16)
where T 0 = 12"trt2[R[(1 + c l a o ] Z / N s ) / V ~ k , U with c being a constant of O(1). The linear temperature dependence of C v is attributed to the constant spinon density of states at low energy region [2] which is justified at low hole density. Since both R and a 0 decrease as the temperature rises, C v / T is an increasing function of T. 7 will be a few mJ tool -~ K 2 for J R [ - 1 when tZ/u ~ 0.05(eV) and increases with increasing applied magnetic field H since ]R[ decreases with increasing H. Let us discuss the resistivity in normal state for Tc(h ) < T < Tc(R ). Hole has charge and moves almost freely. Thus hole carries current. Its lifetime is determined by elastic scattering due to spinons through the hopping. From the Drude like formula for the hole, the resistivity is given by
p=
m*(U/t)ZkB T 47rZn,e2R2h ,
(17)
where the hole density n* is low. If m* is equal to the free electron mass, n* = 102~(1/cm3), and t / U = 0.1, we obtain p = 130/R 2 (ix12 cm) at 100 K. A n o t h e r contribution to the conductivity, associated with, the scattering of the RVB condensed state due to hopping, will be essential as the temperature increases. We consider that the experimental results of the specific heat, C v ~ y T , at low temperature ( T ~ To) and the resistivity, p ~ T, in normal state imply the relation, T c = Tc(h ) and Tc(R ) > Tc(h), as discussed by Anderson et al. [2].
6. Summary The operator transformation developed in this paper is suitable for treating the dynamics of the charged boson (hole) and the spinon, introduced by Anderson, which give an appropriate description for the properties in highly correlated systems. We explicitly showed that the order parameter responsible for superconductivity results when the Bose condensation of the holes and the RVB condensation of pseudofermions (or spinons) occur at the same time. We find that the intersite repulsive interaction gives rise to a marked change of the hole excitation spectrum, which will modify the electron density of states. The long range part of the Coulomb interaction in addion to the short range part will be essential to the understanding of the various properties of high T c oxides. By applying our method to the specific heat at low temperatures and the resistivity in normal state, we obtain the results in agreement with the experimental ones.
References [1] [2] [3] [4] [5] [6] [7] [8]
EW. Anderson, Science 235 (1987) 1196. P.W. Anderson, G. Baskaran, Z. Zou and T. Hsu, Phys. Rev. Lett. 58 (1987) 2790. G. Baskaran and P.W. Anderson, preprint. Y. Isawa, Meeting at the Physical Society of Japan, 1979, unpublished. D.M. Newns, preprint. A.E. Ruckenstein, P.J. Hirschfeld and J. Appel, preprint M. Inui, S. Doniach, EJ. Hirschfeld and A.E. Ruckenstein, preprint. J.E. Hirsch, Phys. Rev. B 35 (1987) 8726.