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Condensate phase fluctuations in a charged Bose gas superlattice
Physica C 152 ( 1988) 289-292 North-Holland, Amsterdam
CONDENSATE PHASE FLUCTUATIONS IN A CHARGED BOSE GAS SUPERLATTICE A. GRIFFIN Department ofPhysics, University of Toronto, Toronto, Ontario, M59lA7, Department of Physics, Kyoto University, Kyoto 606, Japan
Canada *and
Received 30 April 1988
Within the framework of the RVB theory of oxide superconductors, we consider a periodic array of two-dimensional (2D) Bose gases (hoions) with intra-layer and inter-layer Coulomb interactions. Using a mean field approximation consistent with the equivalence of single particle and density fluctuations, we evaluate the single-particle Green’s function in the long wavelength limit. We show that the inter-layer Coulomb interaction is not strong enough to suppress the phase fluctuations sufftciently to stabilize a 2D Bose condensate in each layer. Since we do not consider any hopping between layers, Josephson-type coupling terms are not included.
In the RVB phase, the electrons in the CuO layers in high transition temperature superconductors are described in terms of charge and spin degrees of freedom which are essentially uncoupled [ 1,2]. The charge degrees of freedom are charged Bosons (the holons) and only arise away from the half-filled-band case. Slave Boson formalisms [ 31 are a convenient way of dealing with the holons (as well as the spin degrees of freedom, the so-called spinons). Using a mean field approximation, the superconducting phase is often described as a Bose condensation of holons in the RVB phase [ 3-5 1. Usually these questions are discussed in the context of a Hubbard model with only on-site Coulomb repulsion. However, Isawa, Maekawa and Ebisawa [ 4,6 ] have included a long range Coulomb interaction and argued that this only modifies the holon excitation dispersion relation. The discussion of spinon and holon excitations is usually done for a single CuO layer, ie, for a two-dimensional (2D) Hubbard model. As is well known from the work of Hohenberg [ 7 1, the rigorous Bogoliubov inequality for Bose condensed systems states that the momentum distribution must satisfy n,(T)
n&T
* Permanent address.
1
(1)
This inequality is correct for any dimensionality d and for arbitrary two-particle interactions of the kind v(r-r’ ). Physically, the long wavelength (small q) behavior arises from the condensate phase fluctuations. It is clear [ 7 ] that Sdq( n,) would diverge for dl2 and this means the condensate no( Z’) must vanish. This difficulty is usually ignored under the assumption that the condensate wave function will be stabilized (i.e. the phase fluctuations will be suppressed) through a combination of finite-size effects and coupling with other layers. In the recent literature on oxide superconductors based on a 2D Hubbard model, this coupling is often assumed to be a single holon hopping between neighbouring CuO layers, which effectively makes the holons 3D excitations [ 5,8]. The resulting Bose condensate phase is found not to be very dependent on the details of this inter-layer coupling. On the other hand, Anderson and co-workers [ 91 have emphasized that only “physical” electrons (and not just holons) can hop between CuO layers. Consequently, they argue that superconductivity is associated not with a holon Bose condensate but with holon pairs which are stabilized by a coherent Josephson-type coupling between layers. In the present article, in contrast, we discuss the effect of the long-range Coulomb interaction between holons in a periodic system of CuO layers such
0921-4534/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
290
A. Grifin / Condensate phasefluctuations in a charged Bose gas superlattice
as one finds in all high temperature oxide superconductors. More precisely, our model is that of a Bose “superlattice” of two-dimensional charged Bose gases (ZD-CBG). In contrast to ordinary Fermi superlattices made up of 2D electron gases [ lo], the charge fluctuations and the elementary excitations are essentially the same in a Bose-condensed CBG [ 111. The inter-layer Coulomb interaction leads to a modified Bogoliubov-like spectrum which has a q,-dependence (the layers are in the x-y plane). However, it is apparently not strong enough to stabilize the phase fluctuations of the 2D Bose condensate in each layer. We do not include any single-particle hopping and hence there is no equivalent of the inter-layer Josephson coupling which is often studied in ordinary layered superconductors [ 12 1. In a recent paper [ 111 we discussed the excitation spectrum of a uniform 3D charged Bose gas. It was pointed out that for a dense CBG, one could include the effect of the Bosons outside the condensate in a consistent manner by using the free Bose gas approximation to proper, irreducible quantities. This ensured that the density flucturation spectrum and the single-particle spectrum were the same, a crucial feature of any system with a Bose broken symmetry. One found that apart from the region close to T,, the long wavelength single-particle excitations are given by the usual 3D plasmon dispersion relation involving the total density of Bosons n, even if the condensate is significantly depleted [ no( T) +z n] . In the present paper, we generalize the work of ref. [ 111 to a Bose superlattice, a periodic array of sheets each of which is a ZD-CBG. The Bosons interact via the Coulomb interaction in a given sheet (intra-layer) and between sheets (inter-layer). Mathematically, the problem of solving the real space Dyson-Beliaev equations for the 2 x 2 matrix Green’s function for our coupled system of Bose gas layers is very similar to solving the RPA equation of motion for the density response function in an electron superlattice [ 131. This can already be seen by examining the structure of the Dyson-Beliaev equations in the simple Bogoliubov approximation, as given by eq. (55.26) of ref. [ 141. In general, the solution will involve G&q, -q-G,), where G,= n,2k/d are the 1D reciprocal lattice vectors of the periodic array with the layer spacing d. For our present purposes, it is sufficient to limit ourselves to
the long wavelength limit (qd =x 1 ), in which case one can use the local approximation G&q, -q-G,) “G&q, -4). In this continuum limit, the equations of motion are easily solved and one finds (in the so-called RPA approximation, as described in ref. [ 111
(3) Here we have
m no LX-_0 n p’ 2q,,(4: +42)“2’
cl*0
(4)
and E2=e2 4
+ Qll
&v(q)“W2~ 4P Pl
m
2+ 411
(5)
2’ q=
with 47te2
v(q) = 2 411+qz ’
e =& Qll
zrn 7
2---- 4xe2n Wpl=
m
. (6)
The effective condensate density is no= No/Ad, where No is the number of Bosons with q= 0 in a given sheet of area A (the total density n is defined similarly). The fact that the excitation energy in eq. (5) involves the total density n = no+ ii (even if no <
A. Gr@n / Condensate phase fluctuations in a charged Bose gas superlattice
eqs. ( 4 ) and ( 5 ) . The momentum distribution of the Bosom (holons) is given by [ 141
291
Once again, we see that this will lead to a In ql, divergence in eq. ( 10) and thus the quantum fluctuations destabilize the condensate at both T=O and T# 0.
where No (0) is the Bose distribution. We recall that in a Bose condensate system, the momentum distribution always contains a contribution related to the phase fluctuations of the coherent condensate wavefunction ($3(r)),. This is given by [ 15,161 (%>c=no(T)<@f(q))
*
(8)
The physical content of the relation in eq. ( 1) is that the long wavelength phase fluctuations can lead to the destruction of long range order (Peierls-Landau argument) in systems with dimensionality d< 2. For q,#O, in which case Eq is acoustic, one can use the classical approximation Eq<< kgT and eq. (7) reduces to
(n,> =
%I!+;;‘(‘)
kB
1
T_
4
nomkBT z--_* 4f
1 2’
9ZZO.
(9)
This RPA result is consitent with Bogoliubov’s inequality in eq. ( 1) for a 2D system. The qz dependence disappears and we see that in the expression for the total number of Bosons with q=O,
(10) the q,, integral has the usual logarithmic divergence (In q,, ). This is inadmissable and hence we conclude [7] that no=0 at T#O. At T=O, only the quantum phase fluctuations are left, with ( n,) reducing to V: in eq. (4). In this case, it is seen that for qZ# 0, (n,) does nof have a logarithmic divergence. The q,=O contribution to eq. (7) must be treated separately since Eq=opl. In contrast to the q,# 0 modes, the major contribution is now from the zero point fluctuations
(n,>-
fq,,
+no4!2)
1 -7
2E
4
n0 mapI z---n 29:
1 2’
q,=o.
(11)
We conclude that a satisfactory treatment of a bose condensate of holons in a superlattice of CuO sheets needs to include some electron hopping between layers. It appears that the inter-layer Coulomb interaction is not a symmetry-breaking perturbation for the 2D Bose condensate and our model calculations confirm this. As we have discussed in ref. [ 111 for a 3D-CBG, in the critical region near T,, the dominant poles of the single particle Green’s functions will cease to be those given by eq. (5). In this region, another set of excitations appear, which become the free particle excitations above T, [ 17 1. The detailed spectrum of these critical region excitations has not been worked out, but it seems possible that they will be those of a Bose gas with short-ranged (screened) interactions [ 18 1. Thus, at least in the critical region, the model calculations of ref. [ 8 ] may be appropriate, although further theoretical justification is clearly needed.
Acknowledgements
I would like to thank H. Fukuyama, S. Maekawa, H. Ebisawa and T. Tsuneto for discussion. Prof. Fukuyama first raised the question of whether the inter-layer Coulomb interaction could stabilize the Bose condensate by itself. This work was partially supported by the Japan-Canada Exchange Program, through the Japan Society for the Promotion of Science and NSERC of Canada.
References [ 1] S. Kivelson, D.S. Rokshar and J.P. Sethna, Phys. Rev. B 35 (1987) 8865.
[ 2 ] P.W. Anderson, G. Baskaran, Z. Zou and T. Hsu, Phys. Rev. Lett. 58 (1987) 2790. [3] Z. Zou and P.W. Anderson, Phys. Rev. B 37 ( 1987) 580; A.E. Ruckenstein, P.J. Hirschfeld and J. Appl, Phys. Rev. B 36 (1987) 857. [4] Y. Isawa, S. Maekawa and H. Ebisawa, Physica B 148 (1987) 391.
292
A. Griffin / Condensate phasefluctuations in a charged Bose gas superlattice
[ 5 ] Y. Suzumura, Y. Hasegawa and H. Fukuyama, J. Phys. Sot. Jpn. 57 (1988) 401; H. Fukuyama, Y. Hasegawa and Y. Suzumura, Physica C, to be published. [ 61 S. Maekawa, T. Matsuura, Y. Isawa and H. Ebisawa, PhysicaC152 (1988) 133. [ 71 P.C. Hohenberg, Phys. Rev. 158 ( 1967) 383. [8] X-G. Wen and R. Kan, Phys. Rev. B 37 (1988) 595. [ 91 J.W. Wheatly, T.C. Hsu and P.W. Anderson, Phys. Rev. B 37 (1988) 5897. [lo] A.L. Fetter, Ann. Phys. (N.Y.) 88 (1974) 1. [ 111 A. Griftin, Phys. Rev. B, submitted (Feb., 1988). [ 12 ] R.A. KIemm, A. Luther and M.R. Beasley, Phys. Rev. B 12 (1975) 877.
[ 131 A. Griffin, Phys. Rev. B, submitted (April 1988).
[ 14 ] A.L. Fetter and J.D. Walecka, Quantum Theory of ManyParticle Systems (McGraw-Hill, New York, 197 1). [ 151 R.A. Ferrell, N. Menyhard, H. Schmidt, F. Schwab1 and P. Szepfalusy, Ann. Phys. (N.Y) 47 (1968) 565. [ 161 P.C. Martin, in: Statistical Mechanics at the Turn of the Decade, ed. E.G.D. Cohen (Dekker, New York, 1971) p. 175. [ 171 P. Szepfalusy and I. Kondor, Ann. Phys. (N.Y.) 82 (1974) 82. [ 181 S.-K. Ma, Phys. Rev. Lett. 29 (1972) 1311.
CONDENSATE PHASE FLUCTUATIONS IN A CHARGED BOSE GAS SUPERLATTICE A. GRIFFIN Department ofPhysics, University of Toronto, Toronto, Ontario, M59lA7, Department of Physics, Kyoto University, Kyoto 606, Japan
Canada *and
Received 30 April 1988
Within the framework of the RVB theory of oxide superconductors, we consider a periodic array of two-dimensional (2D) Bose gases (hoions) with intra-layer and inter-layer Coulomb interactions. Using a mean field approximation consistent with the equivalence of single particle and density fluctuations, we evaluate the single-particle Green’s function in the long wavelength limit. We show that the inter-layer Coulomb interaction is not strong enough to suppress the phase fluctuations sufftciently to stabilize a 2D Bose condensate in each layer. Since we do not consider any hopping between layers, Josephson-type coupling terms are not included.
In the RVB phase, the electrons in the CuO layers in high transition temperature superconductors are described in terms of charge and spin degrees of freedom which are essentially uncoupled [ 1,2]. The charge degrees of freedom are charged Bosons (the holons) and only arise away from the half-filled-band case. Slave Boson formalisms [ 31 are a convenient way of dealing with the holons (as well as the spin degrees of freedom, the so-called spinons). Using a mean field approximation, the superconducting phase is often described as a Bose condensation of holons in the RVB phase [ 3-5 1. Usually these questions are discussed in the context of a Hubbard model with only on-site Coulomb repulsion. However, Isawa, Maekawa and Ebisawa [ 4,6 ] have included a long range Coulomb interaction and argued that this only modifies the holon excitation dispersion relation. The discussion of spinon and holon excitations is usually done for a single CuO layer, ie, for a two-dimensional (2D) Hubbard model. As is well known from the work of Hohenberg [ 7 1, the rigorous Bogoliubov inequality for Bose condensed systems states that the momentum distribution must satisfy n,(T)
n&T
* Permanent address.
1
(1)
This inequality is correct for any dimensionality d and for arbitrary two-particle interactions of the kind v(r-r’ ). Physically, the long wavelength (small q) behavior arises from the condensate phase fluctuations. It is clear [ 7 ] that Sdq( n,) would diverge for dl2 and this means the condensate no( Z’) must vanish. This difficulty is usually ignored under the assumption that the condensate wave function will be stabilized (i.e. the phase fluctuations will be suppressed) through a combination of finite-size effects and coupling with other layers. In the recent literature on oxide superconductors based on a 2D Hubbard model, this coupling is often assumed to be a single holon hopping between neighbouring CuO layers, which effectively makes the holons 3D excitations [ 5,8]. The resulting Bose condensate phase is found not to be very dependent on the details of this inter-layer coupling. On the other hand, Anderson and co-workers [ 91 have emphasized that only “physical” electrons (and not just holons) can hop between CuO layers. Consequently, they argue that superconductivity is associated not with a holon Bose condensate but with holon pairs which are stabilized by a coherent Josephson-type coupling between layers. In the present article, in contrast, we discuss the effect of the long-range Coulomb interaction between holons in a periodic system of CuO layers such
0921-4534/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
290
A. Grifin / Condensate phasefluctuations in a charged Bose gas superlattice
as one finds in all high temperature oxide superconductors. More precisely, our model is that of a Bose “superlattice” of two-dimensional charged Bose gases (ZD-CBG). In contrast to ordinary Fermi superlattices made up of 2D electron gases [ lo], the charge fluctuations and the elementary excitations are essentially the same in a Bose-condensed CBG [ 111. The inter-layer Coulomb interaction leads to a modified Bogoliubov-like spectrum which has a q,-dependence (the layers are in the x-y plane). However, it is apparently not strong enough to stabilize the phase fluctuations of the 2D Bose condensate in each layer. We do not include any single-particle hopping and hence there is no equivalent of the inter-layer Josephson coupling which is often studied in ordinary layered superconductors [ 12 1. In a recent paper [ 111 we discussed the excitation spectrum of a uniform 3D charged Bose gas. It was pointed out that for a dense CBG, one could include the effect of the Bosons outside the condensate in a consistent manner by using the free Bose gas approximation to proper, irreducible quantities. This ensured that the density flucturation spectrum and the single-particle spectrum were the same, a crucial feature of any system with a Bose broken symmetry. One found that apart from the region close to T,, the long wavelength single-particle excitations are given by the usual 3D plasmon dispersion relation involving the total density of Bosons n, even if the condensate is significantly depleted [ no( T) +z n] . In the present paper, we generalize the work of ref. [ 111 to a Bose superlattice, a periodic array of sheets each of which is a ZD-CBG. The Bosons interact via the Coulomb interaction in a given sheet (intra-layer) and between sheets (inter-layer). Mathematically, the problem of solving the real space Dyson-Beliaev equations for the 2 x 2 matrix Green’s function for our coupled system of Bose gas layers is very similar to solving the RPA equation of motion for the density response function in an electron superlattice [ 131. This can already be seen by examining the structure of the Dyson-Beliaev equations in the simple Bogoliubov approximation, as given by eq. (55.26) of ref. [ 141. In general, the solution will involve G&q, -q-G,), where G,= n,2k/d are the 1D reciprocal lattice vectors of the periodic array with the layer spacing d. For our present purposes, it is sufficient to limit ourselves to
the long wavelength limit (qd =x 1 ), in which case one can use the local approximation G&q, -q-G,) “G&q, -4). In this continuum limit, the equations of motion are easily solved and one finds (in the so-called RPA approximation, as described in ref. [ 111
(3) Here we have
m no LX-_0 n p’ 2q,,(4: +42)“2’
cl*0
(4)
and E2=e2 4
+ Qll
&v(q)“W2~ 4P Pl
m
2+ 411
(5)
2’ q=
with 47te2
v(q) = 2 411+qz ’
e =& Qll
zrn 7
2---- 4xe2n Wpl=
m
. (6)
The effective condensate density is no= No/Ad, where No is the number of Bosons with q= 0 in a given sheet of area A (the total density n is defined similarly). The fact that the excitation energy in eq. (5) involves the total density n = no+ ii (even if no <
A. Gr@n / Condensate phase fluctuations in a charged Bose gas superlattice
eqs. ( 4 ) and ( 5 ) . The momentum distribution of the Bosom (holons) is given by [ 141
291
Once again, we see that this will lead to a In ql, divergence in eq. ( 10) and thus the quantum fluctuations destabilize the condensate at both T=O and T# 0.
where No (0) is the Bose distribution. We recall that in a Bose condensate system, the momentum distribution always contains a contribution related to the phase fluctuations of the coherent condensate wavefunction ($3(r)),. This is given by [ 15,161 (%>c=no(T)<@f(q))
*
(8)
The physical content of the relation in eq. ( 1) is that the long wavelength phase fluctuations can lead to the destruction of long range order (Peierls-Landau argument) in systems with dimensionality d< 2. For q,#O, in which case Eq is acoustic, one can use the classical approximation Eq<< kgT and eq. (7) reduces to
(n,> =
%I!+;;‘(‘)
kB
1
T_
4
nomkBT z--_* 4f
1 2’
9ZZO.
(9)
This RPA result is consitent with Bogoliubov’s inequality in eq. ( 1) for a 2D system. The qz dependence disappears and we see that in the expression for the total number of Bosons with q=O,
(10) the q,, integral has the usual logarithmic divergence (In q,, ). This is inadmissable and hence we conclude [7] that no=0 at T#O. At T=O, only the quantum phase fluctuations are left, with ( n,) reducing to V: in eq. (4). In this case, it is seen that for qZ# 0, (n,) does nof have a logarithmic divergence. The q,=O contribution to eq. (7) must be treated separately since Eq=opl. In contrast to the q,# 0 modes, the major contribution is now from the zero point fluctuations
(n,>-
fq,,
+no4!2)
1 -7
2E
4
n0 mapI z---n 29:
1 2’
q,=o.
(11)
We conclude that a satisfactory treatment of a bose condensate of holons in a superlattice of CuO sheets needs to include some electron hopping between layers. It appears that the inter-layer Coulomb interaction is not a symmetry-breaking perturbation for the 2D Bose condensate and our model calculations confirm this. As we have discussed in ref. [ 111 for a 3D-CBG, in the critical region near T,, the dominant poles of the single particle Green’s functions will cease to be those given by eq. (5). In this region, another set of excitations appear, which become the free particle excitations above T, [ 17 1. The detailed spectrum of these critical region excitations has not been worked out, but it seems possible that they will be those of a Bose gas with short-ranged (screened) interactions [ 18 1. Thus, at least in the critical region, the model calculations of ref. [ 8 ] may be appropriate, although further theoretical justification is clearly needed.
Acknowledgements
I would like to thank H. Fukuyama, S. Maekawa, H. Ebisawa and T. Tsuneto for discussion. Prof. Fukuyama first raised the question of whether the inter-layer Coulomb interaction could stabilize the Bose condensate by itself. This work was partially supported by the Japan-Canada Exchange Program, through the Japan Society for the Promotion of Science and NSERC of Canada.
References [ 1] S. Kivelson, D.S. Rokshar and J.P. Sethna, Phys. Rev. B 35 (1987) 8865.
[ 2 ] P.W. Anderson, G. Baskaran, Z. Zou and T. Hsu, Phys. Rev. Lett. 58 (1987) 2790. [3] Z. Zou and P.W. Anderson, Phys. Rev. B 37 ( 1987) 580; A.E. Ruckenstein, P.J. Hirschfeld and J. Appl, Phys. Rev. B 36 (1987) 857. [4] Y. Isawa, S. Maekawa and H. Ebisawa, Physica B 148 (1987) 391.
292
A. Griffin / Condensate phasefluctuations in a charged Bose gas superlattice
[ 5 ] Y. Suzumura, Y. Hasegawa and H. Fukuyama, J. Phys. Sot. Jpn. 57 (1988) 401; H. Fukuyama, Y. Hasegawa and Y. Suzumura, Physica C, to be published. [ 61 S. Maekawa, T. Matsuura, Y. Isawa and H. Ebisawa, PhysicaC152 (1988) 133. [ 71 P.C. Hohenberg, Phys. Rev. 158 ( 1967) 383. [8] X-G. Wen and R. Kan, Phys. Rev. B 37 (1988) 595. [ 91 J.W. Wheatly, T.C. Hsu and P.W. Anderson, Phys. Rev. B 37 (1988) 5897. [lo] A.L. Fetter, Ann. Phys. (N.Y.) 88 (1974) 1. [ 111 A. Griftin, Phys. Rev. B, submitted (Feb., 1988). [ 12 ] R.A. KIemm, A. Luther and M.R. Beasley, Phys. Rev. B 12 (1975) 877.
[ 131 A. Griffin, Phys. Rev. B, submitted (April 1988).
[ 14 ] A.L. Fetter and J.D. Walecka, Quantum Theory of ManyParticle Systems (McGraw-Hill, New York, 197 1). [ 151 R.A. Ferrell, N. Menyhard, H. Schmidt, F. Schwab1 and P. Szepfalusy, Ann. Phys. (N.Y) 47 (1968) 565. [ 161 P.C. Martin, in: Statistical Mechanics at the Turn of the Decade, ed. E.G.D. Cohen (Dekker, New York, 1971) p. 175. [ 171 P. Szepfalusy and I. Kondor, Ann. Phys. (N.Y.) 82 (1974) 82. [ 181 S.-K. Ma, Phys. Rev. Lett. 29 (1972) 1311.