- Email: [email protected]
Single-particle green function of the 1D Hubbard model
J. Phys. Chem Solids Vol Primed in Great Britain
54. No. IO. pp Il49-1152.
0022-3697193 56.00 + 0.00 Pergamon Press Ltd
1993
SINGLE-PARTICLE
GREEN FUNCTION
OF THE 11) HUBBARD M. GULACSI
MODEL
and K. S. BEDELL
Theoretical Division, Lor Alsmos National Laboratory, Los Alamos, NM 87545
ABSTRACT
- For the one-dimensional Hubbard model we extended the bosonieation technique, eway from half
filling, in such LI.way that a. general formula is obtained for the eero temperature equal-time single-particle Green function with v&dity over the whole doping rage.
With our method we can calculate, for the first time, the
one-body Green function in both the Tarnonage-Luttinger and Luther-Emery universality classen, to characterize the crossover between these two behaviora and thus, to describe the Mott transition. The method also gives a tool to detennin e the coefficients of the correlation functions which cannot be determined with previously used methods. Keywords: one-dimensional systems, boroni~ation, metal-ins&&or transition
there has been considerable interest in the one-dimensional Hubbard model motivated by various low-dimensional strongly correlated electron systems. In one-dimension, Fermi liquid theory breaks down. The effective low energy theory explicitly exhibits separate elementary excitations carrying charge and spin. Depending on the values of the model parameters, these excitations can have two different types of behavior, corresponding to either the TomonagaRecently,
Luttinger (Tomonage 1950, Luttiagrr 1963, Mattis and Lieb 1965,
D5ydot&bkii
and Larkin l973),
or Luther-
Emery {Luther and Emery 1974, Lee 1975, Emery cl al. 1976, Luther 1976,1977) universality classes. Much attention has been given to the repulsive one-dimensional Hubbard model as a model of a doped Mott insulator and its associated Luttinger liquid behavior. Less attention has been given to the mapping onto the Luther-Emery model and the crossover between these two behaviors. This is our prime focus. From the nonrelativistic fermions, described by ~nihilation and creation operators c;,~ and ci,,, interacting via a repulsive on-site Hubbard term we form Dirac fermions. The latter are described by the annihilation and creation operators \k7,O(z) and !I’$,,(=), with 7 = Ifr representing its spinor component. The normal ordering of the Hubbard interaction term is taken by removing the average charge density of the system. In the normal ordering prescription, the Schriidinger picture operators are given (Ha 1984) as functions of the scalar Bose field ch(z), for which we use the Dell’Antonio (Dell’Antonio et 01. 1972) representrrtion, also known as the Mandelstam representation (Mandelstam 1975), and II(c), the canonical momentum conjugate to G(z). At, this stage, we have two Boson fields (and two canonical momenta) for each (up and down) spin state. While a(z) and II(z) satisfy the equal time a&commutation relation, the chiral components Ok of the scalar field, cP(z) = d o not, satisfy the equal time commutations rules. However, in transforming e+(z) + 9-(z) the spinor components of the Dirac fermions to the chiral components of the scalar bosons the normalization constants are adjusted so that the spinor components satisfy the equal time ~ticommutation relations. We introduce the charge field, as usual, in the form 9, = (9~ + velocity is uz = v&‘, where VF = 2t sin[?r(l - 6)/2] is the Fermi hole doping. In the transformation process, for uy a weak-coupling is obtained. In order to obtain a result which is valid for any values 1149
@‘I)/&. The corresponding velocity, and 5 measures the result, u’,’ = (1 + U/~V~)‘/~ of U, we determine the value
ii50
M. GUL&I andK. S. BEDELL
of uz from the exact Bethe An&z
solution.
The charge sector, with the Boson field and its corresponding canonical momenta m-scaled oppositely by &, has an effective Hamiltonian, up to a multiplicative constant, equivalent to the quantum sine-Gordon theory written in canonical form. Relating the charge channel of the Hubbard chain to the quantum sine-Gordon equation we take the range of the interaction U as the fundamental length unit (Luther 1976). Then, by shifting the latter energy scale with the soliton energy and equating it with the Hubbard charge gap (AU), we obtain U: = l+(?r/2)Av,/m. We calculate the zero temperature equal-time single-particle Green function G~=o(z - y) = (C&Z,,,
+ c~,~c~,~) with the use of a low momenta cutoff Az = AU/ZIP (GuIc& and BeThe asymptotic form of the single-particle Green function, becomes Go(z - y) a a% + (z - y)2 sin[tan-f(~z - y//o& ~p~-(~2/~)g(~~ -_yI)/(WI, where $(I= -_yl) = -b/m/ X/m2 = Sz/uf is the coupling constant of the sine-Gordon theory and oz is the ultraviolet cutoff. In the oz --v 0 limit we obtain the expected form dell 1992).
Go(z
-
y) a
Ix exp[--
Yl
CWd*
Eq. (1) defines the correlation length ((U/t) = $/A r.r,which is identical to the soliton length. This is different from the Anderson localization problem where the correlation length is twice the localization length. From Eq. (1) we see that the Hubbard model at this particular value of doping is a Mott insulator. Away from half filling (6 # 0), the charge degrees of freedom we describe, in a most natural way, by a finite soliton density (doped) quantum sine-Gordon Hamiltonian, i.e., doped Mott insulator. Thus, the massive Thiring model characteristic of the half-filled band case becomes massless upon doping. Or in other words, the doped Hubbard model becomes equivalent to a free massless scalar field (~chuts 1990). The doped quantum sine-Gordon Hamiltonian has already been extensively studied. It describes the well-known soliton lattice structures (H orovite 1981,Pokrovsky and Ttdapov 1979) in onedimension. The soliton lattice is equivalent to the classical two-dimensional sine-Gordon functional with incommensurability along one direction. The electron (hole) concentration is either equivalent to the mean incommensurability of the two-dimensional system or to the soliton (antisoliton~ density of the one-dimensional chain. It is known (Held ant 1982) that the soliton lattice generated by the doped sine-Gordon equation in the classical or semiclassical approach (H orovite 1981) will melt in the fully quantum sine-Gordon problem (Haldam less), giving rise to a Luttinger liquid. Thus, the doped quantum sine-Gordon Hamiltonian becomes massless. The model as described is critical. This description of the doped Hubbard model is different in its starting point from the commonly used bosonization approaches. For example, the massless Thiring model corresponding to the doped Hubbard chain is self-interacting rather than non-intercating. A direct consequence of this fact is the singular behavior of the left and right movers in the U 4 0 limit, which can be of great importance in the study of the two chain problem. The zero temperature equal-time single-particle Green function obtained for 6 # 0 has the wellknown form Gs(x-
x’)
a
[
4 ff: + (z - x’)2
]8/3+1
s
where al is the ultraviolet cutoff, /3 = (u: + l/ IL: - 2)/4 and U: is determined from the exact Bethe Ansatz soltution (R&U and Korcpin 1990).
Green function of the 1D Hubbard model
1151
At the Mott transition, p is an analytic function of 6, with lima+oP --+ l/8 (n&m and Korepin Iwo). Thus, the 6 4 0 limit of Eq. (2) and the momentum distribution will not give the half-filled band result. We demonstarte this result not to be true because the Mott transition is smooth.
The main advantage of using the doped quantum sine-Gordon Hamiltonian is to
overcome inconsistency.
With this we allow, in a natural way, the connection of the half-filled
band with the doped case. From the semi-classical theory of the soliton lattices we know that the presence of the gradient term in the doped sine-Gordon Hamiltoniangenerates a conducting band. In the full quantum problem, this band melts (Hald ane 1982) giving rise to a soliton liquid (i.e. a Luttinger liquid). Depending on the doping states, i.e. soliton or antisolitons, the Luttinger liquid will be formed either at the top or bottom of the soliton gap. The picture presented above is equivalent to the behavior of the charge degrees of freedom of the studied Hubbard model. In which by e.g., hole doping, a Luttinger liquid band of holons opens at the bottom of the half-filled Hubbard gap. The presence of this continuous band is not fatal for the Hubbard gap, which does not collapse. Even if there are spin excitation between the lower and upper Hubbard band, these are not relevant to the present calculation.
U = 8, n = 0.62 ‘\ \ \ 8.937
----
r I I
,’
T-----~!-
_ i,----_ \ \ \,
,’ I-
I I
:
EIp(kll
\ 4.937
_---_---___
/
J-v
_
II
2.343
0.257
Fig. 1: The exact holon band and the firat upper Hubbard (charge soliton) band calculated from the exact Bethe Anssts equations. The holon states are occupied in the ground state up to the holon Fermi points, kp = nw. The holon band, deliminated by the dotted-dashed -e ir empty. The thick dashed curves are the linearised spectrum which we bow&e. The momentum cutoffs corresponding to the lower and upper Hubbard banda are marked by Ai and Aa.
1152
M. Gu~ksr and K. S. BEDELL
The excitations corresponding the half-filled band case. The band (Woynarovich 1982) except gap. The Hubbard gap in the
to the upper Hubbard band are the same (Woynuovieh 1982) a8 in dispersion relation is the same for the upper and lower Hubbard that their energy levels are shifted with the value of the Hubbard doped case is AJ(U) = Au + S’Ar(U) (woynarovich 1982), where
Au is the Hubbard gap for the half-filled band case and A,(V)
= ~~(2~/~)/~~(2~/~~,
with
IV(z) as the Bessel function. Thus, the only way in which we can describe both the half-filled and the doped band of the Hubbard chain, as exhibited by the exact solution, is by considering both the upper and the lower Hubbard bands. We bosonize both bands (see Fig. l.), by defining the lower Hubbard band to be of finite width (GUIscsi and Bed&l 1992), which must vanish as we approach the halffilled band case. This bosonization is accomplished by a four component boson field (Banks et al. 1976, Ha 1984) with two momenta cutoffs. An upper cutoff, Al = 2tb/u: is used to simulate the finite width of the lower Hubbard band and a lower cutoff, A2 = A~(U)/VF is used similarly to the half-filled band case. With the four boson fields defined in this manner, we calculate the zero temperature equal-time single-particle Green function
Gs(t
-
2’) cc
~,Aye-~@~[
4
fx; + (22- E’)2
]@@+l+_I--H]
, 6
where
and Zwancinger D., Phys. Rev. D6 (iS72) 988.
D5yaloshinskii I. Ye. and Larkin A. I., Sov. Phys. JETP S8 (1973) 202. Emery V. J., Luther A. and Peschel I., Phys. Rev. BlS (1976) 1272. Gulaesi M. and BedcU K. S., preprint 1992. Ha Y. K., Phys. Rev. D28 (1984) 1744. Haldane F. D. M., Jour. Phys. Al5
(1982) 507.
Horovits B., Phys. Rev. L&t. 46 (1981) 742. Frahm H. and Korepin V. E., Phys. Rev. B42 (1990) 10553. Lee P. A., Phys. Rev. Lett. 34 (1975) 1247. Luther A. and Emcry V. J., Phya. Rev. Lctt. 53, (1974) 589. Luther A., Phys. Rev. Al4
(1976) 2153 and
B15 (1977) 403.
Luttinger J. M., J. Math. Phys. 4 (1963) 1154. MandeIstam S., Phys. Rev. Dll
(1975) 3026.
Mattis D. C. and Lieb E. H., J. Math. Phys. 6 (1965) 304. Pokrovsky V. L. and Talapov A. L., Phys. Rev. L&t. 42 (1979) 65. Seh& H. J., Phys. Rev. L&t. 64 (1990) 2831. Tomonaga S., Prog. Theor. Phys. 3 (1950) 544.
54. No. IO. pp Il49-1152.
0022-3697193 56.00 + 0.00 Pergamon Press Ltd
1993
SINGLE-PARTICLE
GREEN FUNCTION
OF THE 11) HUBBARD M. GULACSI
MODEL
and K. S. BEDELL
Theoretical Division, Lor Alsmos National Laboratory, Los Alamos, NM 87545
ABSTRACT
- For the one-dimensional Hubbard model we extended the bosonieation technique, eway from half
filling, in such LI.way that a. general formula is obtained for the eero temperature equal-time single-particle Green function with v&dity over the whole doping rage.
With our method we can calculate, for the first time, the
one-body Green function in both the Tarnonage-Luttinger and Luther-Emery universality classen, to characterize the crossover between these two behaviora and thus, to describe the Mott transition. The method also gives a tool to detennin e the coefficients of the correlation functions which cannot be determined with previously used methods. Keywords: one-dimensional systems, boroni~ation, metal-ins&&or transition
there has been considerable interest in the one-dimensional Hubbard model motivated by various low-dimensional strongly correlated electron systems. In one-dimension, Fermi liquid theory breaks down. The effective low energy theory explicitly exhibits separate elementary excitations carrying charge and spin. Depending on the values of the model parameters, these excitations can have two different types of behavior, corresponding to either the TomonagaRecently,
Luttinger (Tomonage 1950, Luttiagrr 1963, Mattis and Lieb 1965,
D5ydot&bkii
and Larkin l973),
or Luther-
Emery {Luther and Emery 1974, Lee 1975, Emery cl al. 1976, Luther 1976,1977) universality classes. Much attention has been given to the repulsive one-dimensional Hubbard model as a model of a doped Mott insulator and its associated Luttinger liquid behavior. Less attention has been given to the mapping onto the Luther-Emery model and the crossover between these two behaviors. This is our prime focus. From the nonrelativistic fermions, described by ~nihilation and creation operators c;,~ and ci,,, interacting via a repulsive on-site Hubbard term we form Dirac fermions. The latter are described by the annihilation and creation operators \k7,O(z) and !I’$,,(=), with 7 = Ifr representing its spinor component. The normal ordering of the Hubbard interaction term is taken by removing the average charge density of the system. In the normal ordering prescription, the Schriidinger picture operators are given (Ha 1984) as functions of the scalar Bose field ch(z), for which we use the Dell’Antonio (Dell’Antonio et 01. 1972) representrrtion, also known as the Mandelstam representation (Mandelstam 1975), and II(c), the canonical momentum conjugate to G(z). At, this stage, we have two Boson fields (and two canonical momenta) for each (up and down) spin state. While a(z) and II(z) satisfy the equal time a&commutation relation, the chiral components Ok of the scalar field, cP(z) = d o not, satisfy the equal time commutations rules. However, in transforming e+(z) + 9-(z) the spinor components of the Dirac fermions to the chiral components of the scalar bosons the normalization constants are adjusted so that the spinor components satisfy the equal time ~ticommutation relations. We introduce the charge field, as usual, in the form 9, = (9~ + velocity is uz = v&‘, where VF = 2t sin[?r(l - 6)/2] is the Fermi hole doping. In the transformation process, for uy a weak-coupling is obtained. In order to obtain a result which is valid for any values 1149
@‘I)/&. The corresponding velocity, and 5 measures the result, u’,’ = (1 + U/~V~)‘/~ of U, we determine the value
ii50
M. GUL&I andK. S. BEDELL
of uz from the exact Bethe An&z
solution.
The charge sector, with the Boson field and its corresponding canonical momenta m-scaled oppositely by &, has an effective Hamiltonian, up to a multiplicative constant, equivalent to the quantum sine-Gordon theory written in canonical form. Relating the charge channel of the Hubbard chain to the quantum sine-Gordon equation we take the range of the interaction U as the fundamental length unit (Luther 1976). Then, by shifting the latter energy scale with the soliton energy and equating it with the Hubbard charge gap (AU), we obtain U: = l+(?r/2)Av,/m. We calculate the zero temperature equal-time single-particle Green function G~=o(z - y) = (C&Z,,,
+ c~,~c~,~) with the use of a low momenta cutoff Az = AU/ZIP (GuIc& and BeThe asymptotic form of the single-particle Green function, becomes Go(z - y) a a% + (z - y)2 sin[tan-f(~z - y//o& ~p~-(~2/~)g(~~ -_yI)/(WI, where $(I= -_yl) = -b/m/ X/m2 = Sz/uf is the coupling constant of the sine-Gordon theory and oz is the ultraviolet cutoff. In the oz --v 0 limit we obtain the expected form dell 1992).
Go(z
-
y) a
Ix exp[--
Yl
CWd*
Eq. (1) defines the correlation length ((U/t) = $/A r.r,which is identical to the soliton length. This is different from the Anderson localization problem where the correlation length is twice the localization length. From Eq. (1) we see that the Hubbard model at this particular value of doping is a Mott insulator. Away from half filling (6 # 0), the charge degrees of freedom we describe, in a most natural way, by a finite soliton density (doped) quantum sine-Gordon Hamiltonian, i.e., doped Mott insulator. Thus, the massive Thiring model characteristic of the half-filled band case becomes massless upon doping. Or in other words, the doped Hubbard model becomes equivalent to a free massless scalar field (~chuts 1990). The doped quantum sine-Gordon Hamiltonian has already been extensively studied. It describes the well-known soliton lattice structures (H orovite 1981,Pokrovsky and Ttdapov 1979) in onedimension. The soliton lattice is equivalent to the classical two-dimensional sine-Gordon functional with incommensurability along one direction. The electron (hole) concentration is either equivalent to the mean incommensurability of the two-dimensional system or to the soliton (antisoliton~ density of the one-dimensional chain. It is known (Held ant 1982) that the soliton lattice generated by the doped sine-Gordon equation in the classical or semiclassical approach (H orovite 1981) will melt in the fully quantum sine-Gordon problem (Haldam less), giving rise to a Luttinger liquid. Thus, the doped quantum sine-Gordon Hamiltonian becomes massless. The model as described is critical. This description of the doped Hubbard model is different in its starting point from the commonly used bosonization approaches. For example, the massless Thiring model corresponding to the doped Hubbard chain is self-interacting rather than non-intercating. A direct consequence of this fact is the singular behavior of the left and right movers in the U 4 0 limit, which can be of great importance in the study of the two chain problem. The zero temperature equal-time single-particle Green function obtained for 6 # 0 has the wellknown form Gs(x-
x’)
a
[
4 ff: + (z - x’)2
]8/3+1
s
where al is the ultraviolet cutoff, /3 = (u: + l/ IL: - 2)/4 and U: is determined from the exact Bethe Ansatz soltution (R&U and Korcpin 1990).
Green function of the 1D Hubbard model
1151
At the Mott transition, p is an analytic function of 6, with lima+oP --+ l/8 (n&m and Korepin Iwo). Thus, the 6 4 0 limit of Eq. (2) and the momentum distribution will not give the half-filled band result. We demonstarte this result not to be true because the Mott transition is smooth.
The main advantage of using the doped quantum sine-Gordon Hamiltonian is to
overcome inconsistency.
With this we allow, in a natural way, the connection of the half-filled
band with the doped case. From the semi-classical theory of the soliton lattices we know that the presence of the gradient term in the doped sine-Gordon Hamiltoniangenerates a conducting band. In the full quantum problem, this band melts (Hald ane 1982) giving rise to a soliton liquid (i.e. a Luttinger liquid). Depending on the doping states, i.e. soliton or antisolitons, the Luttinger liquid will be formed either at the top or bottom of the soliton gap. The picture presented above is equivalent to the behavior of the charge degrees of freedom of the studied Hubbard model. In which by e.g., hole doping, a Luttinger liquid band of holons opens at the bottom of the half-filled Hubbard gap. The presence of this continuous band is not fatal for the Hubbard gap, which does not collapse. Even if there are spin excitation between the lower and upper Hubbard band, these are not relevant to the present calculation.
U = 8, n = 0.62 ‘\ \ \ 8.937
----
r I I
,’
T-----~!-
_ i,----_ \ \ \,
,’ I-
I I
:
EIp(kll
\ 4.937
_---_---___
/
J-v
_
II
2.343
0.257
Fig. 1: The exact holon band and the firat upper Hubbard (charge soliton) band calculated from the exact Bethe Anssts equations. The holon states are occupied in the ground state up to the holon Fermi points, kp = nw. The holon band, deliminated by the dotted-dashed -e ir empty. The thick dashed curves are the linearised spectrum which we bow&e. The momentum cutoffs corresponding to the lower and upper Hubbard banda are marked by Ai and Aa.
1152
M. Gu~ksr and K. S. BEDELL
The excitations corresponding the half-filled band case. The band (Woynarovich 1982) except gap. The Hubbard gap in the
to the upper Hubbard band are the same (Woynuovieh 1982) a8 in dispersion relation is the same for the upper and lower Hubbard that their energy levels are shifted with the value of the Hubbard doped case is AJ(U) = Au + S’Ar(U) (woynarovich 1982), where
Au is the Hubbard gap for the half-filled band case and A,(V)
= ~~(2~/~)/~~(2~/~~,
with
IV(z) as the Bessel function. Thus, the only way in which we can describe both the half-filled and the doped band of the Hubbard chain, as exhibited by the exact solution, is by considering both the upper and the lower Hubbard bands. We bosonize both bands (see Fig. l.), by defining the lower Hubbard band to be of finite width (GUIscsi and Bed&l 1992), which must vanish as we approach the halffilled band case. This bosonization is accomplished by a four component boson field (Banks et al. 1976, Ha 1984) with two momenta cutoffs. An upper cutoff, Al = 2tb/u: is used to simulate the finite width of the lower Hubbard band and a lower cutoff, A2 = A~(U)/VF is used similarly to the half-filled band case. With the four boson fields defined in this manner, we calculate the zero temperature equal-time single-particle Green function
Gs(t
-
2’) cc
~,Aye-~@~[
4
fx; + (22- E’)2
]@@+l+_I--H]
, 6
where
and Zwancinger D., Phys. Rev. D6 (iS72) 988.
D5yaloshinskii I. Ye. and Larkin A. I., Sov. Phys. JETP S8 (1973) 202. Emery V. J., Luther A. and Peschel I., Phys. Rev. BlS (1976) 1272. Gulaesi M. and BedcU K. S., preprint 1992. Ha Y. K., Phys. Rev. D28 (1984) 1744. Haldane F. D. M., Jour. Phys. Al5
(1982) 507.
Horovits B., Phys. Rev. L&t. 46 (1981) 742. Frahm H. and Korepin V. E., Phys. Rev. B42 (1990) 10553. Lee P. A., Phys. Rev. Lett. 34 (1975) 1247. Luther A. and Emcry V. J., Phya. Rev. Lctt. 53, (1974) 589. Luther A., Phys. Rev. Al4
(1976) 2153 and
B15 (1977) 403.
Luttinger J. M., J. Math. Phys. 4 (1963) 1154. MandeIstam S., Phys. Rev. Dll
(1975) 3026.
Mattis D. C. and Lieb E. H., J. Math. Phys. 6 (1965) 304. Pokrovsky V. L. and Talapov A. L., Phys. Rev. L&t. 42 (1979) 65. Seh& H. J., Phys. Rev. L&t. 64 (1990) 2831. Tomonaga S., Prog. Theor. Phys. 3 (1950) 544.