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Optical conductivity of the two-dimensional Hubbard model
PHYSICA
Physica B 186-188 (1993) 959-961 North-Holland
Optical conductivity of the two-dimensional Hubbard model W. B r e n i g lnstitut fiir Theoretische Physik, Universitiit zu K61n, Germany We analyze the optical conductivity of the two-dimensional Hubbard model using the projection technique. The current self-energy and the static current susceptibility are evaluated perturbatively applying a novel real space summation technique. Results for the Drude weight and the incoherent spectrum at various filling densities are presented.
1. Introduction
magnetic current density and the inverse mass tensor, respectively.
The dynamical electromagnetic response serves as an important tool for the investigation of correlated electronic systems. Recent experimental interest has focused in particular on the optical conductivity of the high-temperature superconductors [1]. The theoretical insight into this quantity for two-dimensional systems is limited to finite-size numerical calculations [2]. In this work we study the weak coupling properties of the dynamical conductivity using the,projection method in Liouville space as an alternative and analytic approach.
2. Optical conductivity To obtain the dynamical linear response of the current we introduce a time-dependent vector potential A(r, t) into the Hubbard model by means of Peierls' description. Up to second order in A and for wavelengths large compared with the lattice constant, this results in
H(A) = - t
j,=it
t [ ( r , - r , , ) ClCmo
~
l~m]
,
(2.2) too -- t
~
[(r, - rmL(r, - rm)S~Cm~ + h . c . l .
(I,m),~r
Within linear response theory the optical conductivity can be written as o'~t3(o~) = e
I
j~
Jr3 . . . . io+ '
where ( . . [ . . ) is Mori's scalar product and L denotes the Liouville operator defined by LO = [H, O] . At zero magnetic field the optical conductivity and the grand canonical average of the inverse tensor are diagonal with only one independent element, denoted by cr(z) and (7.). Using Mori-Zwanzig projection technique we reformulate the conductivity as [3]: o-(z) = ie 2
(7")
z - M(z)(1/(7")) ' 1
M(z) = ( L v j
~'~ {c~.cm¢ +h.c.}
+
( l , m ) ,a
Lvj ) ,
(2.4)
e2 E r~A,~A~ . (2.1) The first two terms are the Hubbard model in the usual notation, and the last two terms represent the coupling to the vector potential. The electric field is given by E = i o J A and j~ and %~ are the para-
Correspondence to: W. Brenig, Institut fiir Theoretische Physik, Universit~it zu K61n, ZiilpicherstraBe 77, W-5000 KBln 41, Germany.
Q=I-lj)
1 ~-~ (Jl.
The total oscillator strength determined by the f-sum rule results in (7"). The Drude weight D of the zero-frequency delta peak in the real part of the optical conductivity is given by e21r(j, I PJ~ ), where P projects onto the non-ergodic sector of the Liouville space. The time derivative L v j of the current is
Lvj = -iUt
~_~
* I~n i~(Cl_~Cl+~x_ ~ + c*t+~x-~Ct-~).
I,~,be--± 1
{)921-4526/93/$06.00 (~) 1993 - Elsevier Science Publishers B.V. All rights reserved
(2.5)
W. Brenig / Optical conductivity of the 2D Hubbard model
960
The index l + / z x implies a site displaced b y / x lattice constants into the x-direction relative to the site r t. These relations are exact. In order to proceed we approximate the current self-energy and the average kinetic energy in the n u m e r a t o r to second order in U. In this case M(z) is
r~ z -
3
- z M ° ( z ) = [X°vs(z) - xRvj(0)]
where X°vj(z) is obtained from the analytic continuation of the Fourier transform of the imaginary time, propagator { 7", {(Luj)+ ( A ) L v j ( 0 ) } ) o, where ( . . . ) 0 is the grand canonical average with respect to the noninteracting ground state, T , { . . . } is the time-ordering operator and A ~ [0, /3] is the imaginary time. This propagator can be calculated using standard diagram techniques. However an evaluation of the resulting graphs in k-space is limited by the high dimensionality of the m o m e n t u m integrals. Further complications arise from the geometry of U m k l a p p processes. In view of this, we express the X 0L u / ( l W • n) susceptibility through a real-space lattice sum as X;~vjCio.,,,) =
g
(2.6)
0.1
8
8 - -
14
/L 12.
~0.05
5
10
15
Fig. 1. Convergence of the lattice sum. Labels refer to /It,,,. 14 implies that Gt(z ) with At ~< 5 have to be taken into account. In the zero-temperature limit the second-order correction to kinetic energy can be evaluated in a similar fashion from a linked cluster expansion observing that 2(r)lr_ o = Fir_,,( H v ) ] r o- We find Ato t =
4U2t2T U
x
~'.
[B,(iw,+iwt)B,+~+,_,x(-iwa)
(~)2-(~>o-- - - -
2
T~[B,,(i%+iw,) 4
/,~l
l.bt,v,m 1
+ 2B,+.+ , ( i % + iwt)B,
(2.9)
× Bu(-iogt)]i=+in.r+ o •
+ Bu(iw . + iw,)Bt+(~_~>x+(-iwl)
,_~.(-i~l)l, (2.7)
where % = 2narT, T is the temperature and the quantities B~m(iw) are the symmetrized particle-hole propagators.
Using our results for the memory function and the kinetic energy at zero temperature we can extract the D r u d e weight: 1 2
~re
D =2
/(
O Re[MR(w)]
1
==0& )
Ow
(2.10)
T Btm(i(o., ) = - ~ ~ [G,Cie. + iw+)Gm(ie.) + l++m], en
I
(2.8)
where e, = (2n + 1)IrT and G , ( i e , ) is the two-dimensional lattice Green function for a given spin direction. After performing the frequency sums we are left with two one-dimensional integrals, for each term in the lattice sum, which we have calculated numerically at zero temperature [3]. All lattice Green functions are generated from elliptic integrals. The remaining sum over sites converges within a few steps. The n u m b e r of sites necessary to include increases away from half filling. Figure 1 illustrates this for the case of the imaginary part of the retarded A%us(w) R at a filling of n = 0.4. Each graph displays the change of the lattice sum upon inclusion of the next contribution in Atot, which is the sum of nearest-neighbor steps relative to the origin, associated with the sites l, m, p and q,
'
'
'
I
'
'
'
I
'
'
'
I
'
'
0.8-
I
'
'
'
o
o-
.4P+" . .......... O ............. • O.e
,~
0.4
- - -
U=O
O.Z /
* U-2
/ 0
,
,
t
I 0.2
,
,
,
I
,
a ,
0.4
I 0.6
,
,
,
I
,
,
,
0,8
1"1
Fig. 2. Drude weight as a function of filling for U/t =2 (T=O, t = l ) .
W. Brenig / Optical conductivity of the 2D Hubbard model the
U=2.0
961
optical conductivity at various densities for
U/t = 2. The two insets refer to the complete range of o
0.8 0.03 r.'---i
~
0
5
10
0
1
6
to
15
r---n
o.0 %
0.02
3 0.4 ~
I~ 0.01
0
0.2
2
4
6
0
2
4
6
Fig. 3. Incoherent optical conductivity for various band fillings (T = 0, t = 1). This is shown in fig. 2 for U = 2 as a function of the density. The solid line represents the noninteracting Drude weight. The open circles display the reduction of D resulting from the loss of kinetic energy, while the full circles include the renormalization due to MR(to). Even though in fig. 2 the ratio of U to the bare electron bandwidth is 1/4, the relative size of the reduction in D is small, i.e. pertu~bation theory seems applicable at intermediate densities. Close to a metalinsualtor transition at n = 1 our weak correlation approach is inconclusive regarding the form of D(n, U). Figure 3 shows the incoherent contribution (r~c(to) to
nonzero spectral intensity of Mr~(to). For to--> 0 the real part of the incoherent optical conductivity levels off at a constant, while the imaginary part is proportional to 1/to. The maximum in Re[trRc(to)] as a function of to for some values of n is due to a zero in the real part of the memory function and not to the particular value of U/t. To obtain separate structure in gRc(to ) similar to an upper Hubbard band, one has to apply values of U/t ~>8. This is beyond the perturbative approach presented here. This work was supported by the Sonderforschungsbereich 341 of the Deutsche Forschungsgemeinschaft.
References
[1] T. Timusk, M. Reedyk, R. Hughes, D.A. Bonn, J.D. Garret, J.E. Greedan, C.V. Stager, D.B. Tanner, Feng Gao, S.L. Herr, K. Kamarfis, G.A. Thomas, S.L. Cooper, J. Orenstein, L.F. Schneemeyer and A.J. Millis, Physica C 162-164 (1989) 841; Z. Schlesinger, R.T. Collins, F. Holtzberg, C. Field, G. Koen and A. Gupta, Phys. Rev. B 41 (1990) 11237. [2] A. Moreo and E. Dagotto, Phys. Rev. B 42 (1990) 4786; W. Stephan and P. Horsch, Phys. Rev. B 42 (1990) 8736; E. Dagotto, A. Moreo, F. Ortolani, J. Riera and D.J. Scalapino, Phys. Rev. B 45 (1992) 7311. [3] W. Brenig, Z. Phys. B 89 (1992) 187.
Physica B 186-188 (1993) 959-961 North-Holland
Optical conductivity of the two-dimensional Hubbard model W. B r e n i g lnstitut fiir Theoretische Physik, Universitiit zu K61n, Germany We analyze the optical conductivity of the two-dimensional Hubbard model using the projection technique. The current self-energy and the static current susceptibility are evaluated perturbatively applying a novel real space summation technique. Results for the Drude weight and the incoherent spectrum at various filling densities are presented.
1. Introduction
magnetic current density and the inverse mass tensor, respectively.
The dynamical electromagnetic response serves as an important tool for the investigation of correlated electronic systems. Recent experimental interest has focused in particular on the optical conductivity of the high-temperature superconductors [1]. The theoretical insight into this quantity for two-dimensional systems is limited to finite-size numerical calculations [2]. In this work we study the weak coupling properties of the dynamical conductivity using the,projection method in Liouville space as an alternative and analytic approach.
2. Optical conductivity To obtain the dynamical linear response of the current we introduce a time-dependent vector potential A(r, t) into the Hubbard model by means of Peierls' description. Up to second order in A and for wavelengths large compared with the lattice constant, this results in
H(A) = - t
j,=it
t [ ( r , - r , , ) ClCmo
~
l~m]
,
(2.2) too -- t
~
[(r, - rmL(r, - rm)S~Cm~ + h . c . l .
(I,m),~r
Within linear response theory the optical conductivity can be written as o'~t3(o~) = e
I
j~
Jr3 . . . . io+ '
where ( . . [ . . ) is Mori's scalar product and L denotes the Liouville operator defined by LO = [H, O] . At zero magnetic field the optical conductivity and the grand canonical average of the inverse tensor are diagonal with only one independent element, denoted by cr(z) and (7.). Using Mori-Zwanzig projection technique we reformulate the conductivity as [3]: o-(z) = ie 2
(7")
z - M(z)(1/(7")) ' 1
M(z) = ( L v j
~'~ {c~.cm¢ +h.c.}
+
( l , m ) ,a
Lvj ) ,
(2.4)
e2 E r~A,~A~ . (2.1) The first two terms are the Hubbard model in the usual notation, and the last two terms represent the coupling to the vector potential. The electric field is given by E = i o J A and j~ and %~ are the para-
Correspondence to: W. Brenig, Institut fiir Theoretische Physik, Universit~it zu K61n, ZiilpicherstraBe 77, W-5000 KBln 41, Germany.
Q=I-lj)
1 ~-~ (Jl.
The total oscillator strength determined by the f-sum rule results in (7"). The Drude weight D of the zero-frequency delta peak in the real part of the optical conductivity is given by e21r(j, I PJ~ ), where P projects onto the non-ergodic sector of the Liouville space. The time derivative L v j of the current is
Lvj = -iUt
~_~
* I~n i~(Cl_~Cl+~x_ ~ + c*t+~x-~Ct-~).
I,~,be--± 1
{)921-4526/93/$06.00 (~) 1993 - Elsevier Science Publishers B.V. All rights reserved
(2.5)
W. Brenig / Optical conductivity of the 2D Hubbard model
960
The index l + / z x implies a site displaced b y / x lattice constants into the x-direction relative to the site r t. These relations are exact. In order to proceed we approximate the current self-energy and the average kinetic energy in the n u m e r a t o r to second order in U. In this case M(z) is
r~ z -
3
- z M ° ( z ) = [X°vs(z) - xRvj(0)]
where X°vj(z) is obtained from the analytic continuation of the Fourier transform of the imaginary time, propagator { 7", {(Luj)+ ( A ) L v j ( 0 ) } ) o, where ( . . . ) 0 is the grand canonical average with respect to the noninteracting ground state, T , { . . . } is the time-ordering operator and A ~ [0, /3] is the imaginary time. This propagator can be calculated using standard diagram techniques. However an evaluation of the resulting graphs in k-space is limited by the high dimensionality of the m o m e n t u m integrals. Further complications arise from the geometry of U m k l a p p processes. In view of this, we express the X 0L u / ( l W • n) susceptibility through a real-space lattice sum as X;~vjCio.,,,) =
g
(2.6)
0.1
8
8 - -
14
/L 12.
~0.05
5
10
15
Fig. 1. Convergence of the lattice sum. Labels refer to /It,,,. 14 implies that Gt(z ) with At ~< 5 have to be taken into account. In the zero-temperature limit the second-order correction to kinetic energy can be evaluated in a similar fashion from a linked cluster expansion observing that 2(r)lr_ o = Fir_,,( H v ) ] r o- We find Ato t =
4U2t2T U
x
~'.
[B,(iw,+iwt)B,+~+,_,x(-iwa)
(~)2-(~>o-- - - -
2
T~[B,,(i%+iw,) 4
/,~l
l.bt,v,m 1
+ 2B,+.+ , ( i % + iwt)B,
(2.9)
× Bu(-iogt)]i=+in.r+ o •
+ Bu(iw . + iw,)Bt+(~_~>x+(-iwl)
,_~.(-i~l)l, (2.7)
where % = 2narT, T is the temperature and the quantities B~m(iw) are the symmetrized particle-hole propagators.
Using our results for the memory function and the kinetic energy at zero temperature we can extract the D r u d e weight: 1 2
~re
D =
/(
O Re[MR(w)]
1
==0& )
Ow
(2.10)
T Btm(i(o., ) = - ~ ~ [G,Cie. + iw+)Gm(ie.) + l++m], en
I
(2.8)
where e, = (2n + 1)IrT and G , ( i e , ) is the two-dimensional lattice Green function for a given spin direction. After performing the frequency sums we are left with two one-dimensional integrals, for each term in the lattice sum, which we have calculated numerically at zero temperature [3]. All lattice Green functions are generated from elliptic integrals. The remaining sum over sites converges within a few steps. The n u m b e r of sites necessary to include increases away from half filling. Figure 1 illustrates this for the case of the imaginary part of the retarded A%us(w) R at a filling of n = 0.4. Each graph displays the change of the lattice sum upon inclusion of the next contribution in Atot, which is the sum of nearest-neighbor steps relative to the origin, associated with the sites l, m, p and q,
'
'
'
I
'
'
'
I
'
'
'
I
'
'
0.8-
I
'
'
'
o
o-
.4P+" . .......... O ............. • O.e
,~
0.4
- - -
U=O
O.Z /
* U-2
/ 0
,
,
t
I 0.2
,
,
,
I
,
a ,
0.4
I 0.6
,
,
,
I
,
,
,
0,8
1"1
Fig. 2. Drude weight as a function of filling for U/t =2 (T=O, t = l ) .
W. Brenig / Optical conductivity of the 2D Hubbard model the
U=2.0
961
optical conductivity at various densities for
U/t = 2. The two insets refer to the complete range of o
0.8 0.03 r.'---i
~
0
5
10
0
1
6
to
15
r---n
o.0 %
0.02
3 0.4 ~
I~ 0.01
0
0.2
2
4
6
0
2
4
6
Fig. 3. Incoherent optical conductivity for various band fillings (T = 0, t = 1). This is shown in fig. 2 for U = 2 as a function of the density. The solid line represents the noninteracting Drude weight. The open circles display the reduction of D resulting from the loss of kinetic energy, while the full circles include the renormalization due to MR(to). Even though in fig. 2 the ratio of U to the bare electron bandwidth is 1/4, the relative size of the reduction in D is small, i.e. pertu~bation theory seems applicable at intermediate densities. Close to a metalinsualtor transition at n = 1 our weak correlation approach is inconclusive regarding the form of D(n, U). Figure 3 shows the incoherent contribution (r~c(to) to
nonzero spectral intensity of Mr~(to). For to--> 0 the real part of the incoherent optical conductivity levels off at a constant, while the imaginary part is proportional to 1/to. The maximum in Re[trRc(to)] as a function of to for some values of n is due to a zero in the real part of the memory function and not to the particular value of U/t. To obtain separate structure in gRc(to ) similar to an upper Hubbard band, one has to apply values of U/t ~>8. This is beyond the perturbative approach presented here. This work was supported by the Sonderforschungsbereich 341 of the Deutsche Forschungsgemeinschaft.
References
[1] T. Timusk, M. Reedyk, R. Hughes, D.A. Bonn, J.D. Garret, J.E. Greedan, C.V. Stager, D.B. Tanner, Feng Gao, S.L. Herr, K. Kamarfis, G.A. Thomas, S.L. Cooper, J. Orenstein, L.F. Schneemeyer and A.J. Millis, Physica C 162-164 (1989) 841; Z. Schlesinger, R.T. Collins, F. Holtzberg, C. Field, G. Koen and A. Gupta, Phys. Rev. B 41 (1990) 11237. [2] A. Moreo and E. Dagotto, Phys. Rev. B 42 (1990) 4786; W. Stephan and P. Horsch, Phys. Rev. B 42 (1990) 8736; E. Dagotto, A. Moreo, F. Ortolani, J. Riera and D.J. Scalapino, Phys. Rev. B 45 (1992) 7311. [3] W. Brenig, Z. Phys. B 89 (1992) 187.