- Email: [email protected]
Electron–hole resonant states in the d=∞ Hubbard model
Physica B 312–313 (2002) 522–524
Electron–hole resonant states in the d=N Hubbard model$ V. Janis$* Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Praha, Czech Republic
Abstract Mean-field limit (d=N) of the Hubbard model at half filling is investigated within the parquet approach. Simplified parquet equations are explicitly solved at intermediate coupling where nonself-consistent approximations of the FLEX type become numerically unstable. A new solution with anomalous vertex functions (complex effective interaction) was found that is interpreted as a state containing resonant electron–hole pairs. r 2002 Elsevier Science B.V. All rights reserved. PACS: 71.10.Fd; 71.30.+h Keywords: Resonant pair states; Simplified parquet equations; Complex effective interaction
Detailed understanding of the physics in the transition region between weak and strong coupling remains still abstruse. Two-particle functions may develop divergences connected with various phase transitions. Such a situation demands application of nonperturbative methods with nontrivial vertex corrections. One of the most advanced nonperturbative two-particle approximations is the parquet approach combining summation of Feynman diagrams with equations of motion for one and two-particle Green functions [1]. The parquet approximation sums multiple scatterings from different two-particle channels in a self-consistent manner so that the output of the Bethe–Salpeter equation from one channel is part of the input to the Bethe–Salpeter equations in the other scattering channels. The principal new qualitative (exact) feature of the parquet equations is that only integrable singularities are allowed in the two-particle functions. Thereby unphysical and spurious poles are suppressed. The author recently proposed a simplification of the parquet equations in the critical region of the zero-
$
The work was supported in part by a collaborative Grant ME-383 between the Ministry of Education, Youth and Sports of the Czech Republic and the National Science Foundation of the USA. *Tel.: +42-2-66052153; fax: +42-2-86890527. E-mail address: [email protected] (V. Jani$s).
temperature metal–insulator transition [1]. It takes into account only diagrams potentially divergent at the transition point and keeps the relevant variables of the vertex functions in which the singularity is expected to appear. This leads to two coupled equations for the irreducible vertex functions Leh and LU from the electron–hole and interaction channels, respectively. If we further simplify the absolute terms in the Bethe– Salpeter equations we can decouple the equations to Ref. [2]: Leh ðqÞ ¼ U ; 1 /ðU=1 þ /Leh Gm Gk SÞGm Gm SðqÞ/U=ð1 þ /Leh Gm Gk SÞGk Gk SðqÞ
ð1aÞ LU ðqÞ ¼ U ; 1 þ /U=ð1 /LU Gm Gm S /LU Gk Gk SÞGm Gk SðqÞ ð1bÞ where we used abbreviations for a convolution of two one-particle propagators with a vertex G 1 X GðkÞGs ðkÞGs0 ðk þ qÞ ð2Þ /GGs Gs0 SðqÞ ¼ bN k and q ¼ ðk; ioÞ for four-momenta. These equations are a generalization of the FLEX equations with a dynamical
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 3 3 0 - 8
V. Jani$s / Physica B 312–313 (2002) 522–524 U=2 4
parquet FLEX
3.5 3
Re Γ
vertex renormalization. If we put Leh ¼ LU ¼ 0 on the r.h.s of Eqs. (1a) and (1b) we recover the single-channel (FLEX) approximations for the full two-particle vertex from the electron–hole and interaction channels [3]. We have to complete the parquet approximation with an equation for the self-energy. The self-energy correction to the Hartree static term is derived from the Schwinger–Dyson equation with the full vertex G ¼ Leh þ LU U: It can be written within this approximate scheme with the above abbreviations as U X Ss ðkÞ ¼ Gs ðk þ qÞ ½ðLeh ðqÞ UÞ bN q
523
2.5 2 1.5 1 -4
-2
0
2
4
ω
/Gs Gs SðqÞ þ Gs ðk þ qÞ
) /UGm Gk SðqÞ : 1 þ /U=ð1 /LU Gm Gm S /LU Gk Gk SÞGm Gk SðqÞ
Fig. 1. Real part of the normal two-particle vertex function in the phase with resonant pair states compared with the FLEX result.
ð3Þ Eqs. ð1aÞ–ð3Þ define our approximation completely. These equations retain the restriction on the integrability of singularities in two-particle functions. For explicit numerical calculations, we resort to the spin and charge symmetric case (half filling) of an infinite dimensional Bethe lattice with a semi-elliptic density of states. The four-vectors reduce to frequencies. For a qualitative assessment of the solution we put Leh ¼ U and LU ¼ LU ð0Þ in (1) and (3). We hence keep from the parquet equations only Eq. ð1Þ at the Fermi energy o ¼ 0: This approximation does not change the (non)integrability of singularities in the vertex function. The only parameter to be determined from the simplified parquet equations is the effective interaction LU ð0Þ: This interaction is real at weak coupling LU ð0ÞEU: However, when the interaction is strong enough, UEw; w is half-bandwidth, we find that two real solutions of (3) coalesce and split into the complex plane. We then obtain two complex conjugate solutions for the effective interaction LU ð0Þ: The complex phase of the effective interaction means that the number of quasiparticles for each of the complex solutions is not conserved. This fact may seem unphysical but has a rather natural explanation. When we approach the metal–insulator transition electron–hole pairs start to bind and at some threshold resonant states emerge. It means that electron–hole bound states are created and exist on a macroscopically long but finite time scale before they are eventually annihilated. There are no bound states on the infinite time scale and hence no divergence in the equilibrium two-particle vertex function. Quasiparticles from the asymptotic space (Bloch waves) can be absorbed to create a temporary bound electron–hole pair (Im LU ð0Þo0) and the particles from the electron–hole resonant states can be emitted back to the asymptotic space (Im LU ð0Þ > 0). Since the whole
system of particles must conserve the energy, the absorption and emission processes must be in equilibrium. It means that both complex solutions to the parquet equations (the full two-particle vertex G) must contribute to the one-particle self-energy with the same probability. The solutions with a complex effective interaction can be viewed upon as new phases breaking a symmetry of the weakly coupled state. The broken symmetry can be defined only in the two-particle space. At weak coupling the two-particle vertex fulfills a reflection symmetry in the complex energy plane GðzÞ ¼ Gn ðzn Þ saying that its complex structure is entirely generated by the complex energy.1 It is just this symmetry that breaks in the state with resonant pairs, since a new complex structure is added by the effective interaction from the parquet equations. The new phase hence introduces anomalous twoparticle functions. We can define normal and anomalous vertex functions as follows: Gnorm ðzÞ ¼ 12 ðGðzÞ þ Gn ðzn ÞÞ; Ganom ðzÞ ¼ 12 ðGðzÞ Gn ðzn ÞÞ:
ð4Þ
The normal part preserves the weak-coupling reflection symmetry while the anomalous one represents ‘‘order parameters’’ measuring the impact of the existence of resonant states. Since the contributions to the selfenergy must be balanced, only the normal part Gnorm appears in (3) so that the quasiparticles have infinite lifetime at the Fermi energy. Fig. 1 shows the impact of the existence of a complex effective interaction on the form of the full two-particle 1
Note that the reflection symmetry does not stand for the electron–hole symmetry. The latter is characterized by GðzÞ ¼ GðzÞ and is always satisfied.
V. Jani$s / Physica B 312–313 (2002) 522–524
524
vertex ReGnorm ðoÞ compared to the FLEX approximation. The central peak is suppressed and the weight is shifted from the Fermi energy to its neighborhood. The anomalous vertex function Ganom ðoÞ (order parameter) is plotted in Fig. 2.
U=2 0.8
Re Γanom Im Γanom
0.6 0.4 0.2 0
References
-0.2 -0.4 -0.6 -4
-2
0
2
4
ω Fig. 2. Real and imaginary parts of the anomalous two-particle vertex function.
[1] V. Jani$s, Phys. Rev. B 60 (1999) 11 345. [2] V. Jani$s, in: J. Bon$ca, et al., (Eds.), Open Problems in Strongly Correlated Electron Systems, Kluwer Academic, Dordrecht, 2001, p. 361. [3] N.E. Bickers, D.J. Scalapino, Ann. Phys. (NY) 193 (1989) 206.
Electron–hole resonant states in the d=N Hubbard model$ V. Janis$* Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Praha, Czech Republic
Abstract Mean-field limit (d=N) of the Hubbard model at half filling is investigated within the parquet approach. Simplified parquet equations are explicitly solved at intermediate coupling where nonself-consistent approximations of the FLEX type become numerically unstable. A new solution with anomalous vertex functions (complex effective interaction) was found that is interpreted as a state containing resonant electron–hole pairs. r 2002 Elsevier Science B.V. All rights reserved. PACS: 71.10.Fd; 71.30.+h Keywords: Resonant pair states; Simplified parquet equations; Complex effective interaction
Detailed understanding of the physics in the transition region between weak and strong coupling remains still abstruse. Two-particle functions may develop divergences connected with various phase transitions. Such a situation demands application of nonperturbative methods with nontrivial vertex corrections. One of the most advanced nonperturbative two-particle approximations is the parquet approach combining summation of Feynman diagrams with equations of motion for one and two-particle Green functions [1]. The parquet approximation sums multiple scatterings from different two-particle channels in a self-consistent manner so that the output of the Bethe–Salpeter equation from one channel is part of the input to the Bethe–Salpeter equations in the other scattering channels. The principal new qualitative (exact) feature of the parquet equations is that only integrable singularities are allowed in the two-particle functions. Thereby unphysical and spurious poles are suppressed. The author recently proposed a simplification of the parquet equations in the critical region of the zero-
$
The work was supported in part by a collaborative Grant ME-383 between the Ministry of Education, Youth and Sports of the Czech Republic and the National Science Foundation of the USA. *Tel.: +42-2-66052153; fax: +42-2-86890527. E-mail address: [email protected] (V. Jani$s).
temperature metal–insulator transition [1]. It takes into account only diagrams potentially divergent at the transition point and keeps the relevant variables of the vertex functions in which the singularity is expected to appear. This leads to two coupled equations for the irreducible vertex functions Leh and LU from the electron–hole and interaction channels, respectively. If we further simplify the absolute terms in the Bethe– Salpeter equations we can decouple the equations to Ref. [2]: Leh ðqÞ ¼ U ; 1 /ðU=1 þ /Leh Gm Gk SÞGm Gm SðqÞ/U=ð1 þ /Leh Gm Gk SÞGk Gk SðqÞ
ð1aÞ LU ðqÞ ¼ U ; 1 þ /U=ð1 /LU Gm Gm S /LU Gk Gk SÞGm Gk SðqÞ ð1bÞ where we used abbreviations for a convolution of two one-particle propagators with a vertex G 1 X GðkÞGs ðkÞGs0 ðk þ qÞ ð2Þ /GGs Gs0 SðqÞ ¼ bN k and q ¼ ðk; ioÞ for four-momenta. These equations are a generalization of the FLEX equations with a dynamical
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 3 3 0 - 8
V. Jani$s / Physica B 312–313 (2002) 522–524 U=2 4
parquet FLEX
3.5 3
Re Γ
vertex renormalization. If we put Leh ¼ LU ¼ 0 on the r.h.s of Eqs. (1a) and (1b) we recover the single-channel (FLEX) approximations for the full two-particle vertex from the electron–hole and interaction channels [3]. We have to complete the parquet approximation with an equation for the self-energy. The self-energy correction to the Hartree static term is derived from the Schwinger–Dyson equation with the full vertex G ¼ Leh þ LU U: It can be written within this approximate scheme with the above abbreviations as U X Ss ðkÞ ¼ Gs ðk þ qÞ ½ðLeh ðqÞ UÞ bN q
523
2.5 2 1.5 1 -4
-2
0
2
4
ω
/Gs Gs SðqÞ þ Gs ðk þ qÞ
) /UGm Gk SðqÞ : 1 þ /U=ð1 /LU Gm Gm S /LU Gk Gk SÞGm Gk SðqÞ
Fig. 1. Real part of the normal two-particle vertex function in the phase with resonant pair states compared with the FLEX result.
ð3Þ Eqs. ð1aÞ–ð3Þ define our approximation completely. These equations retain the restriction on the integrability of singularities in two-particle functions. For explicit numerical calculations, we resort to the spin and charge symmetric case (half filling) of an infinite dimensional Bethe lattice with a semi-elliptic density of states. The four-vectors reduce to frequencies. For a qualitative assessment of the solution we put Leh ¼ U and LU ¼ LU ð0Þ in (1) and (3). We hence keep from the parquet equations only Eq. ð1Þ at the Fermi energy o ¼ 0: This approximation does not change the (non)integrability of singularities in the vertex function. The only parameter to be determined from the simplified parquet equations is the effective interaction LU ð0Þ: This interaction is real at weak coupling LU ð0ÞEU: However, when the interaction is strong enough, UEw; w is half-bandwidth, we find that two real solutions of (3) coalesce and split into the complex plane. We then obtain two complex conjugate solutions for the effective interaction LU ð0Þ: The complex phase of the effective interaction means that the number of quasiparticles for each of the complex solutions is not conserved. This fact may seem unphysical but has a rather natural explanation. When we approach the metal–insulator transition electron–hole pairs start to bind and at some threshold resonant states emerge. It means that electron–hole bound states are created and exist on a macroscopically long but finite time scale before they are eventually annihilated. There are no bound states on the infinite time scale and hence no divergence in the equilibrium two-particle vertex function. Quasiparticles from the asymptotic space (Bloch waves) can be absorbed to create a temporary bound electron–hole pair (Im LU ð0Þo0) and the particles from the electron–hole resonant states can be emitted back to the asymptotic space (Im LU ð0Þ > 0). Since the whole
system of particles must conserve the energy, the absorption and emission processes must be in equilibrium. It means that both complex solutions to the parquet equations (the full two-particle vertex G) must contribute to the one-particle self-energy with the same probability. The solutions with a complex effective interaction can be viewed upon as new phases breaking a symmetry of the weakly coupled state. The broken symmetry can be defined only in the two-particle space. At weak coupling the two-particle vertex fulfills a reflection symmetry in the complex energy plane GðzÞ ¼ Gn ðzn Þ saying that its complex structure is entirely generated by the complex energy.1 It is just this symmetry that breaks in the state with resonant pairs, since a new complex structure is added by the effective interaction from the parquet equations. The new phase hence introduces anomalous twoparticle functions. We can define normal and anomalous vertex functions as follows: Gnorm ðzÞ ¼ 12 ðGðzÞ þ Gn ðzn ÞÞ; Ganom ðzÞ ¼ 12 ðGðzÞ Gn ðzn ÞÞ:
ð4Þ
The normal part preserves the weak-coupling reflection symmetry while the anomalous one represents ‘‘order parameters’’ measuring the impact of the existence of resonant states. Since the contributions to the selfenergy must be balanced, only the normal part Gnorm appears in (3) so that the quasiparticles have infinite lifetime at the Fermi energy. Fig. 1 shows the impact of the existence of a complex effective interaction on the form of the full two-particle 1
Note that the reflection symmetry does not stand for the electron–hole symmetry. The latter is characterized by GðzÞ ¼ GðzÞ and is always satisfied.
V. Jani$s / Physica B 312–313 (2002) 522–524
524
vertex ReGnorm ðoÞ compared to the FLEX approximation. The central peak is suppressed and the weight is shifted from the Fermi energy to its neighborhood. The anomalous vertex function Ganom ðoÞ (order parameter) is plotted in Fig. 2.
U=2 0.8
Re Γanom Im Γanom
0.6 0.4 0.2 0
References
-0.2 -0.4 -0.6 -4
-2
0
2
4
ω Fig. 2. Real and imaginary parts of the anomalous two-particle vertex function.
[1] V. Jani$s, Phys. Rev. B 60 (1999) 11 345. [2] V. Jani$s, in: J. Bon$ca, et al., (Eds.), Open Problems in Strongly Correlated Electron Systems, Kluwer Academic, Dordrecht, 2001, p. 361. [3] N.E. Bickers, D.J. Scalapino, Ann. Phys. (NY) 193 (1989) 206.