- Email: [email protected]
Time-dependent Gutzwiller theory of pair fluctuations in the Hubbard model
Physica C 460–462 (2007) 1041–1042 www.elsevier.com/locate/physc
Time-dependent Gutzwiller theory of pair fluctuations in the Hubbard model Falk Gu¨nther *, Go¨tz Seibold Brandenburg Technical University, P.O. Box 101344, 03013 Cottbus, Germany Available online 28 March 2007
Abstract The time-dependent Gutzwiller approximation (TDGA) is extended towards the inclusion of pair correlations. As a first application and in order to check the quality of the method we evaluate superconducting phase transitions in the attractive Hubbard model. It turns out that the TDGA can capture the crossover from weak to strong coupling and that the temperature scale of the intermediate regime is in good agreement with Quantum Monte Carlo calculations in contrast to the BCS approximation. In case of particle–hole symmetry we show the consistency of the approach from the degeneracy of both particle–particle- and particle–hole-channels. Ó 2007 Elsevier B.V. All rights reserved. PACS: 71.10.Fd; 71.10.Li; 74.20.z Keywords: Attractive Hubbard model; Gutzwiller approximation; Pairing instabilities
The attractive Hubbard model represents a generic Hamiltonian in order to study the crossover from weak coupling BCS superconductivity to superfluidity of charge bosons (for a review see e.g. Ref. [1]). Motivated by this problem, earlier work based on the Gutzwiller approximation (GA) or related slave-boson approaches [2–6] has investigated superconducting (SC) properties in the ordered phase. In this paper we develop a scheme within the time-dependent GA (TDGA), which allows to approach the SC phase from the disordered regime. Starting point is the charge-rotational invariant Gutzwiller free energy functional for the Hubbard model: X X z F GA ¼ tij hWþ ð1 2J zi Þ i Ai s Aj Wj i l i;j
þU
X
i
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Di J zi 1 þ tan2 ðui Þ ;
ð1Þ
i
where tij denotes the hopping parameter between sites i and j. The Nambu vectors Wi ¼ ð^ci" ; ^ci# Þ contain the local spin*
Corresponding author. E-mail address: [email protected] (F. Gu¨nther).
0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.210
cir Þ. r-electron creation (destruction) operators ^cþ ir ð^ z J zi ¼ 12 Wþ i s Wi is the z-component of the pseudo-spin vector and sz denotes the Pauli matrix. The matrix Ai describes a local rotation in charge space by the angle ui so that a superconducting order can be established, but also renormalizes the kinetic energy. The chemical potential l is introduced to conserve the particle number. In order to investigate the response of the system to a small external perturbation we expand Eq. (1) in the generalized density matrix and double occupancy deviations. The free energy expansion in momentum space reads as follows: 1X 0 F GA F GA V dJ þ 0 þ trfh dqg þ q dJ q q N |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} dF pp
dqq 1 X dqq þ ; ðMq Þ q dT N dT q q |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð2Þ
dF ph
where we restricted to paramagnetic saddle-point solutions in the normal state hJ i i0 ¼ 0 so that particle–particle- (pp) and particle–hole- (ph) channel terms decouple. The first
1042
F. Gu¨nther, G. Seibold / Physica C 460–462 (2007) 1041–1042 0.5
Localization
11 Q
-1.0
1 0 -1 -2
-1.5 -2.0 0
2
kBTc/B
hc lattice kT=0.0
-0.5
4
6
0
8
2 U/B 6
0
10
1
vph ðx; qÞ ¼ ½1 þ v0ph ðx; qÞMq v0ph ðx; qÞ; vpp ðx; qÞ ¼ ½1 v0pp ðx; qÞV 1 v0pp ðx; qÞ;
ð3Þ
where v0pp=ph denotes the non-interacting susceptibilities obtained within the GA. We now apply the approach developed above to the investigation of the pp- and ph-instabilities for an infinite dimensional (hyper cubic) attractive Hubbard model restricting to half filling. In case of the ph-channel the (charge density wave, CDW) instability occurs at q = Q = [p, p,. . .] and only the [1,1]-entry of the matrix MQ does not vanish. We find that in this special case of particle–hole symmetry the pp- and ph-effective interactions are related by 2M11Q = V with: U ; j8e0 j
0
1
2
3
4
5
6
7
|U|/B
term in Eq. (2) is the saddle-point free energy and tr{h0dq} contains the single-particle excitations on the GA level. In the particle–hole term dFph we have already antiadiabatically eliminated the deviation of the double occupancy parameter dDi [7] and kept the part of the density matrix that commutes with the total particle number. Note that dFph couples the local density fluctuations dq Pq with inter-site charge fluctuations of the form dT q ¼ kr ðek þ ekþq Þnkr . The particle–particle term dFpp describes Gaussian fluctuations of the superconducting order parameter dhJ i i. For both channels the interaction kernels V(U) (in case of pp) and Mq(U) (in case of ph) depend on the momentum q and the bare interaction U. Note that within our notation Hartree–Fock theory (HF) yields U in the pp- and U/2 in the ph-channel. Having proceeded so far we can now evaluate the ph- and pp-susceptibilities on the RPA level obtaining:
u¼
Localization
0.2 0.1
-6 -4 -2
Fig. 1. The effective ph/pp-interaction at zero temperature versus jUj/B. B is the bandwidth. The inset shows the complete range of the interaction for positive and negative U from localization to the Brinkman–Rice transition.
ð1 þ uÞ ; ð1 uÞ
hc lattice n=1.0
0.3
n=1
|U|/B
V ¼ 4e0 ðu 2Þ
TDGA: ph, pp QMC BCS
0.4
n=1.0 U
V, M
V, M11 Q
0
ð4Þ
Fig. 2. Critical temperature versus the on-site interaction jUj/B. B is the bandwidth. The kBTc to pp- and ph-instability lines are degenerated.
which proves the consistency of our charge-rotational invariant TDGA. In Fig. 1 the U-dependence of M11Q and V at zero temperature is shown. For small attractions the effective interactions approach the limit of HF theory (dashed line). At a critical negative U/B 6.5 one has a transition towards localized pairs at which the interactions vanish. In contrast one observes a divergence in M11Q and V for U > 0 at the Brinkman–Rice transition (see the inset to Fig. 1). Finally we calculate the transition temperatures for the phase transitions towards SC and CDW order from Eq. (3) again for a half filled hc lattice. In Fig. 2 the resulting critical temperature as a function of jUj/B is shown and compared with the results from BCS theory and Quantum Monte Carlo calculations (QMC) from Ref. [8]. Summary and conclusion The charge-rotational invariant TDGA was used to calculate the stability of SC and CDW phases. Since the GA becomes exact in infinite dimensions we found a good agreement with the QMC data even for the order of magnitude of T max c . In contrast to the BCS theory the TDGA qualitatively captures at least the crossover from weak to strong coupling. References [1] R. Micnas, J. Ranninger, S. Robaszkiewicz, Rev. Mod. Phys. 62 (1990) 113. [2] B.R. Bulka, Phys. Stat. Sol. B 180 (1993) 401. [3] B.R. Bulka, S. Robaszkiewcz, Phys. Rev. B 54 (1996) 13138. [4] J.O. Sofo, C.A. Balseiro, Phys. Rev. 45 (1992) 377. [5] R. Fre´sard, P. Wo¨lfle, Int. J. Mod. Phys. B 6 (1992) 237. [6] M. Bak, R. Micnas, J. Phys.: Condens. Matter 10 (1998) 9029. [7] G. Seibold, J. Lorenzana, Phys. Rev. Lett. 86 (2001) 2605. [8] M. Jarrell, Phys. Rev. Lett. 69 (1998) 168.
Time-dependent Gutzwiller theory of pair fluctuations in the Hubbard model Falk Gu¨nther *, Go¨tz Seibold Brandenburg Technical University, P.O. Box 101344, 03013 Cottbus, Germany Available online 28 March 2007
Abstract The time-dependent Gutzwiller approximation (TDGA) is extended towards the inclusion of pair correlations. As a first application and in order to check the quality of the method we evaluate superconducting phase transitions in the attractive Hubbard model. It turns out that the TDGA can capture the crossover from weak to strong coupling and that the temperature scale of the intermediate regime is in good agreement with Quantum Monte Carlo calculations in contrast to the BCS approximation. In case of particle–hole symmetry we show the consistency of the approach from the degeneracy of both particle–particle- and particle–hole-channels. Ó 2007 Elsevier B.V. All rights reserved. PACS: 71.10.Fd; 71.10.Li; 74.20.z Keywords: Attractive Hubbard model; Gutzwiller approximation; Pairing instabilities
The attractive Hubbard model represents a generic Hamiltonian in order to study the crossover from weak coupling BCS superconductivity to superfluidity of charge bosons (for a review see e.g. Ref. [1]). Motivated by this problem, earlier work based on the Gutzwiller approximation (GA) or related slave-boson approaches [2–6] has investigated superconducting (SC) properties in the ordered phase. In this paper we develop a scheme within the time-dependent GA (TDGA), which allows to approach the SC phase from the disordered regime. Starting point is the charge-rotational invariant Gutzwiller free energy functional for the Hubbard model: X X z F GA ¼ tij hWþ ð1 2J zi Þ i Ai s Aj Wj i l i;j
þU
X
i
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Di J zi 1 þ tan2 ðui Þ ;
ð1Þ
i
where tij denotes the hopping parameter between sites i and j. The Nambu vectors Wi ¼ ð^ci" ; ^ci# Þ contain the local spin*
Corresponding author. E-mail address: [email protected] (F. Gu¨nther).
0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.210
cir Þ. r-electron creation (destruction) operators ^cþ ir ð^ z J zi ¼ 12 Wþ i s Wi is the z-component of the pseudo-spin vector and sz denotes the Pauli matrix. The matrix Ai describes a local rotation in charge space by the angle ui so that a superconducting order can be established, but also renormalizes the kinetic energy. The chemical potential l is introduced to conserve the particle number. In order to investigate the response of the system to a small external perturbation we expand Eq. (1) in the generalized density matrix and double occupancy deviations. The free energy expansion in momentum space reads as follows: 1X 0 F GA F GA V dJ þ 0 þ trfh dqg þ q dJ q q N |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} dF pp
dqq 1 X dqq þ ; ðMq Þ q dT N dT q q |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð2Þ
dF ph
where we restricted to paramagnetic saddle-point solutions in the normal state hJ i i0 ¼ 0 so that particle–particle- (pp) and particle–hole- (ph) channel terms decouple. The first
1042
F. Gu¨nther, G. Seibold / Physica C 460–462 (2007) 1041–1042 0.5
Localization
11 Q
-1.0
1 0 -1 -2
-1.5 -2.0 0
2
kBTc/B
hc lattice kT=0.0
-0.5
4
6
0
8
2 U/B 6
0
10
1
vph ðx; qÞ ¼ ½1 þ v0ph ðx; qÞMq v0ph ðx; qÞ; vpp ðx; qÞ ¼ ½1 v0pp ðx; qÞV 1 v0pp ðx; qÞ;
ð3Þ
where v0pp=ph denotes the non-interacting susceptibilities obtained within the GA. We now apply the approach developed above to the investigation of the pp- and ph-instabilities for an infinite dimensional (hyper cubic) attractive Hubbard model restricting to half filling. In case of the ph-channel the (charge density wave, CDW) instability occurs at q = Q = [p, p,. . .] and only the [1,1]-entry of the matrix MQ does not vanish. We find that in this special case of particle–hole symmetry the pp- and ph-effective interactions are related by 2M11Q = V with: U ; j8e0 j
0
1
2
3
4
5
6
7
|U|/B
term in Eq. (2) is the saddle-point free energy and tr{h0dq} contains the single-particle excitations on the GA level. In the particle–hole term dFph we have already antiadiabatically eliminated the deviation of the double occupancy parameter dDi [7] and kept the part of the density matrix that commutes with the total particle number. Note that dFph couples the local density fluctuations dq Pq with inter-site charge fluctuations of the form dT q ¼ kr ðek þ ekþq Þnkr . The particle–particle term dFpp describes Gaussian fluctuations of the superconducting order parameter dhJ i i. For both channels the interaction kernels V(U) (in case of pp) and Mq(U) (in case of ph) depend on the momentum q and the bare interaction U. Note that within our notation Hartree–Fock theory (HF) yields U in the pp- and U/2 in the ph-channel. Having proceeded so far we can now evaluate the ph- and pp-susceptibilities on the RPA level obtaining:
u¼
Localization
0.2 0.1
-6 -4 -2
Fig. 1. The effective ph/pp-interaction at zero temperature versus jUj/B. B is the bandwidth. The inset shows the complete range of the interaction for positive and negative U from localization to the Brinkman–Rice transition.
ð1 þ uÞ ; ð1 uÞ
hc lattice n=1.0
0.3
n=1
|U|/B
V ¼ 4e0 ðu 2Þ
TDGA: ph, pp QMC BCS
0.4
n=1.0 U
V, M
V, M11 Q
0
ð4Þ
Fig. 2. Critical temperature versus the on-site interaction jUj/B. B is the bandwidth. The kBTc to pp- and ph-instability lines are degenerated.
which proves the consistency of our charge-rotational invariant TDGA. In Fig. 1 the U-dependence of M11Q and V at zero temperature is shown. For small attractions the effective interactions approach the limit of HF theory (dashed line). At a critical negative U/B 6.5 one has a transition towards localized pairs at which the interactions vanish. In contrast one observes a divergence in M11Q and V for U > 0 at the Brinkman–Rice transition (see the inset to Fig. 1). Finally we calculate the transition temperatures for the phase transitions towards SC and CDW order from Eq. (3) again for a half filled hc lattice. In Fig. 2 the resulting critical temperature as a function of jUj/B is shown and compared with the results from BCS theory and Quantum Monte Carlo calculations (QMC) from Ref. [8]. Summary and conclusion The charge-rotational invariant TDGA was used to calculate the stability of SC and CDW phases. Since the GA becomes exact in infinite dimensions we found a good agreement with the QMC data even for the order of magnitude of T max c . In contrast to the BCS theory the TDGA qualitatively captures at least the crossover from weak to strong coupling. References [1] R. Micnas, J. Ranninger, S. Robaszkiewicz, Rev. Mod. Phys. 62 (1990) 113. [2] B.R. Bulka, Phys. Stat. Sol. B 180 (1993) 401. [3] B.R. Bulka, S. Robaszkiewcz, Phys. Rev. B 54 (1996) 13138. [4] J.O. Sofo, C.A. Balseiro, Phys. Rev. 45 (1992) 377. [5] R. Fre´sard, P. Wo¨lfle, Int. J. Mod. Phys. B 6 (1992) 237. [6] M. Bak, R. Micnas, J. Phys.: Condens. Matter 10 (1998) 9029. [7] G. Seibold, J. Lorenzana, Phys. Rev. Lett. 86 (2001) 2605. [8] M. Jarrell, Phys. Rev. Lett. 69 (1998) 168.