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Hall response of strongly correlated electrons
PHYSiCA® ELSEVIER
Physica C 341-348 (2000) 129-132 www elsevier.nl/Iocate/physc
Hall response of strongly correlated electrons P. Prelovgek, D. Veberie, a M. Long b and X. Zotos c ~J. Stefan Institute, SI-1001 Ljubljana, Slovenia b D e p a r t m e n t of Physics, University of Birmingham, Edgbaston, B i r m i n g h a m B15 2TT, United Kingdom CInstitut R o m a n d de Recherche Num@rique en Physique des Mat@riaux ( E P F L - P P H ) , CH-1015 Lausanne, Switzerland Theoretical understanding and the calculation of the Hall response of strongly correlated electrons is addressed. It is shown that the Hall constant RH in a tight-binding model of correlated electrons on a ladder at T = 0 can be expressed in terms of derivatives of the ground state energy with respect to external magnetic and electric fields. This novel method is used for the analysis of the t-J model on finite size ladders. It is found that for a single hole /~H is hole-like and close to the semiclassical value. In odd-leg ladders, RH behaves quite regularly changing sign as a function of doping, the variation being quantitatively close to experimental results in cuprates. The study is extended to general reactive Hall response and consequantly to quasi-lD systems where RH can be simply related to 1-D charge stiffness of the system.
1. I N T R O D U C T I O N The proper u n d e r s t a n d i n g of the Hall effect in systems of strongly correlated electrons is still a challenge for theoreticians. The efforts have been m o t i v a t e d by the physics of cuprates, superconducting at high t e m p e r a t u r e , viewed as doped M o t t - H u b b a r d insulators. Measurements of the Hall constant RH(T) in the normal metallic state of cuprates reveal a sign change with doping and besides t h a t an anomalously strong t e m p e r a t u r e dependence [1,2]. Investigations of the Hall effect in models of strongly correlated electrons [3-5] resulted in quite controversial conclusions. The dynamical Hall constant /)H (a~) has been studied within the linear response t h e o r y for the t - J and H u b b a r d model. Results are quite consistent for the highfrequency q u a n t i t y R~I = /~H(CO ~ oo) [4], showing a transition from a hole-like R~ > 0 to an electron-like R~ < 0 at a finite doping. For the most interesting d.c. limit RH = /~H(w = 0), however, some results o b t a i n e d for 2D systems at low doping and T -+ 0 indicate even RH < 0 [5], instead of the expected hole-like semiclassical be0921-4534/00/$ - see fi'ont matter (© 2000 Elsevier Science B.M Pll S0921-4534(00)00413-5
havior. Recently the l a t t e r result has been recovered analytically for a single hole doped into a 2D A F M [6]. Here we study the T = 0 Hall response as a ground state p r o p e r t y of a tight-binding model of strongly correlated electrons. We present two alternative ways to the d e t e r m i n a t i o n of the d.c. RH. One is by approaching the system with a ladder geometry, where a finite transverse width avoids the p r o b l e m of low-energy relaxation mechanism [7,8]. A n o t h e r is to investigate more general reactive Hall response in proper higherdimensional system, where again RH can be expressed with g r o u n d - s t a t e quantities only. This proves particularly meaningful approach to quasione dimensional systems, where the Hall constant appears to be related solely to the easy-axis stiffness. 2. L A D D E R
SYSTEMS
We first analyze a tight-binding model of interacting fermions on a ladder geometry with M legs in the y direction and L rungs in the x direction with periodic b o u n d a r y conditions [8]. The All rights reserved.
lq PrelovSek et al./Physica C 341-348 (2000) 129-132
130
following ingredients are needed in the model in order to study the Hall response: a) a finite transverse electric field $ y = A/e0, b) a homogeneous magnetic field B perpendicular to the ladder, introduced via the Peierls substitution into the hopping integrals. Within the Landau gauge A = B ( - y , 0) only the phases of hopping in the .r direction are modified as ~,, = (m - ~ ) ~ = (m - rh)e0B, c) a steady electric current density j* = j induced by piercing a closed ladder in the y direction with a flux, modifying the hopping term by a phase O. So the relevant terms of the model are
H = H~
Hy + Ha + Hint,
+
(1)
M
Hx = - t
Z rn=l
Z
~ i ( ~ - ° ) ' 3k C m , i + l , s C~m i s -H.c.),(2) -r
is
M-1
:-t'
*
z m=l
HA = A E ( m
+ H.c.),
(a)
is
-- fft)nrni,
(4)
im
where the interaction term Hint is not dependent on introduced fields. The idea is to study the ground state energy E of the system as a function of A, 0, ~ in order to evaluate the Hall constant RH, R. -
$ jB
A j~
(5)
In the absence of the magnetic field, ~ = 0, the starting point is an equilibrium and nonpolar ground state with the current density j c( OE/O0 = 0 and the polarization P c( OE/OA = 0. Such a ground state might correspond in finitesize ladder systems to 0o # 0, whereas it follows Ao = 0 by symmetry. To simulate the Hall effect we analyze systems with small current j # 0 imposed by a finite deviation 0 = 00 + 0, and with a magnetic field ~ # 0. At the same time we choose A # 0 so that the system remains nonpolar, P = 0. Evaluating A s induced by 0 and related j, we arrive at the result RH =
N/~°/,~ 0 o ' eo EA z~Eo o
(6)
E A0 : E o0a a E0
=
(7)
where only the derivatives of the ground state energy E(0, A, ~) evaluated at the equilibrium enter, and N = L M . It can be shown that the expression is equivalent to the one obtained via the linear response theory for dynamic RH(W) evaluated at w -----0,
-~H (w) =
I B
ayx (w)
(8) '
where aaZ(w) are dynamic conductivities. The new approach has nevertheless some clear advantages relative to the linear response theory approach: a) only the knowledge of only the ground state energy is required, b) the condition for j = 0, P = 0 in the reference ground state of finite size systems, being crucial in small systems, is more transparent. The method can be easily tested for noninteracting electrons on a two-leg (M = 2) ladder. It can be shown that expected semiclassical results are recovered: a) RH = --1/neeo for an empty upper band where ne is the density of electrons, and b) RH = 1/nheo for a filled lower band where nh = 2 - - ne is the density of holes in the upper band. As an open problem we study the Hall constant RH of correlated electrons within the isotropic t-J model, where
Hint=JE4 4,,
(9)
(iY) and J is equal along the rungs and legs, as well as t' = t. E(O, A, qo) are calculated in finite size ladders via the Lanczos diagonalization technique. A notrivial case now is already that of a single hole Nh = 1 on ladders with different M. Note that one would in this situation expect RH = N/eo, since we are deling with a single positively charged carrier (quasipartiele) in a system. Consistent although systematically higher value is found in M = 2 leg ladder for most J / t studied, while the semicalssical result seems to be approached in M > 2 ladders. It should be here pointed out that in finite-size s y s t e m s / ~ a v difo fers significantly from Eo~v, so that for J < t both quantities can even be of a different sign.
f~
Prelov~eket al./Plo,sica C 341-348 (2000) 129-132
20.0 i
.
i x
,
J=0,4t . - - RH*
.
o
10.0 ',,Xx x "..
c°~
O5X3 o6x3 x LSCO
o
x
0.0
-10.0
-20.0
131
easy to perform. Also, for D > 1 one would like to study systems with periodic boundary conditions where it is not possible to impose infinitesimally small homogeneous magnetic field B. An alternative is however to study the reactive w ~ 0 Hall response RH = /~H(W = 0) for a magnetic field modulated in the y direction, i.e. ~m = R e ( - i e ° B e i q m ) , q
0.0
0.2
0.4
0.6
0.8
n~
Figure 1. Hall constant eoRH vs. hole doping nh for the three-leg ladder and J = 0.4 t. b-~ll line serves as a guide to the eye. Dashed line represents the R~I result [3]. Shown are also experimental results for LSCO taken from Ref. [2].
(10)
Assuming a homogeneous current j* such a field induces a modulated electric field and potential in the y direction, i.e. we have to consider A --+ Aq. It should be noted that in such a situation at T = 0 the electric field $ y and E* are out of phase, as is the case of j* and g~, characteristic for a reactive response of as first considered for many-body systems by Kohn [9], At w ~ 0 we can identify,
2leg =
a~=(w ~ 0) = - - D o ,
(11)
O2
Results for more holes are less uniform. E.g. for two holes, N h 2, on a M = 2 ladder we get RH < 0 for a large J/t regime, consistent with previous studies [7]. This evident deviation from the semiclassical result seems to be related to the existence of the spin gap and the bound state of two holes on such a system. On the other hand, for odd leg (e.g. M = 3) ladder again results consistent with the semiclassical result are recovered. Moreover in M = 3 ladder we can follow the dependence of RH on hole doping nh with the result shown in Fig. 1. One can speculate that the M = 3 ladder already simulates well the behavior of a planar lattice. In fact our result agree quite reasonable with the experimental ones for RH in the normal state of prototype cuprate, La2_,Sr=CuO4 [2] (also presented in Fig. 1), where the planar hole concentration is directly related to x. =
3. REACTIVE HALL RESPONSE Previous approach cannot be extended directly to the 2D (or more D) systems, since the limit M -+ oo is ambiguous as well numerically not
aq~(w --~ 0) - ae~ Ofiq q20Aq '
a~x(~o--+ O) -
(12)
e2 a2E° q OAqO0
2Beg OD~ q2 OAq'
(13)
where fiq is the electron density modulated in the y direction, and D~, D~ are charge stifnesses (Drude weights), ^
( 0 ] - T~,qlO)
D 0~, q
--
2
-Re ~
-
(14)
(Oij~lrn)(rnlj~"[O),
m¢O
£rn
-- E0
defined for uniform and modulated currents respectively. Finally we get RH
=
--
10D~
q~o eOD~ Ofiq lira
~
10D~
e°D~ On
,
j~,q, (15)
assuming that q --+ 0 limit is smooth. The expression is appealing since it gives a direct, intuitive understanding of the change of sign of charge carriers in the vicinity of a metalinsulator transition. It e.g. immediately yields a semiclassical results both in the low-doping limit where D~ cx c~ and RH ~ 1/eoCh.
132
t? Prelovgek et al./Physica C 341-348 (2000) 129-132
4. 20
N
6
o N=2x7 * N=2x8 ,', N=2xl 0
10
......
o ....
6. 7.
o~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . •
O
•
@@
5.
O
8. -10
-20 0
.J/I=2. . . . . . . . 0.2
0.4
0.6
9. 0.8
Figure 2. RH vs. hole doping nh for weakly coupled t-J chains with J = 2t, calculated via the analytical relation (full line) and for various twoleg ladders with a interchain hopping t~/t = 0.5.
4. Q U A S I - 1 D S Y S T E M S The result (15) is particularly useful for a quasi1D system where D~ -~ D, i.e. the charge stiffness becomes the 1D easy axis one, since the weak interchain coupling reduces the coherence in the y-direction. We test the expression on the solvable 1D t-J model with J = 2t, where D(n) can be expressed analytically. Numerically we calculate R h for a M = 2 ladder, where now the coupling between legs is weak, i.e. only via the interchain hopping t ~ < t. Results shown in Fig. 2 seem to confirm the relation, although some scatter in numerical data is evident and can be attributed to finite-size effects. REFERENCES 1. For a review see e . g . N . P . Ong, in Physical
Properties of High Temperature Superconductors, ed. by D. M. Ginsberg (World Scientific, 2. 3.
Singapore, 1990), Vol. 2. H.Y. Hwang, et al., Phys. Rev. Lett. 72, 2636 (1994). H . E . Castillo and C. A. Balseiro, Phys. Rev. Lett. 68, 121 (1992).
B. S. Shastry, B. I. Shraiman, and R. R. P. Singh, Phys. Rev. Lett. 70, 2004 (1993). F . F . Assaad and M. Imada, Phys. Rev. Lett. 74, 3868 (1995). P. Prelov~ek, Phys. Rev. B 55, 9219 (1997). H. Tsunetsugu and M. Imada, J. Phys. Soc. Jpn. 66, 1876 (1997). P. Prelov~ek, M. Long, T. Marke~, and X. Zotos, Phys. Rev. Lett. 83, 2785 (1999). W. Kohn, Phys. Rev. 133, A171 (1964).
Physica C 341-348 (2000) 129-132 www elsevier.nl/Iocate/physc
Hall response of strongly correlated electrons P. Prelovgek, D. Veberie, a M. Long b and X. Zotos c ~J. Stefan Institute, SI-1001 Ljubljana, Slovenia b D e p a r t m e n t of Physics, University of Birmingham, Edgbaston, B i r m i n g h a m B15 2TT, United Kingdom CInstitut R o m a n d de Recherche Num@rique en Physique des Mat@riaux ( E P F L - P P H ) , CH-1015 Lausanne, Switzerland Theoretical understanding and the calculation of the Hall response of strongly correlated electrons is addressed. It is shown that the Hall constant RH in a tight-binding model of correlated electrons on a ladder at T = 0 can be expressed in terms of derivatives of the ground state energy with respect to external magnetic and electric fields. This novel method is used for the analysis of the t-J model on finite size ladders. It is found that for a single hole /~H is hole-like and close to the semiclassical value. In odd-leg ladders, RH behaves quite regularly changing sign as a function of doping, the variation being quantitatively close to experimental results in cuprates. The study is extended to general reactive Hall response and consequantly to quasi-lD systems where RH can be simply related to 1-D charge stiffness of the system.
1. I N T R O D U C T I O N The proper u n d e r s t a n d i n g of the Hall effect in systems of strongly correlated electrons is still a challenge for theoreticians. The efforts have been m o t i v a t e d by the physics of cuprates, superconducting at high t e m p e r a t u r e , viewed as doped M o t t - H u b b a r d insulators. Measurements of the Hall constant RH(T) in the normal metallic state of cuprates reveal a sign change with doping and besides t h a t an anomalously strong t e m p e r a t u r e dependence [1,2]. Investigations of the Hall effect in models of strongly correlated electrons [3-5] resulted in quite controversial conclusions. The dynamical Hall constant /)H (a~) has been studied within the linear response t h e o r y for the t - J and H u b b a r d model. Results are quite consistent for the highfrequency q u a n t i t y R~I = /~H(CO ~ oo) [4], showing a transition from a hole-like R~ > 0 to an electron-like R~ < 0 at a finite doping. For the most interesting d.c. limit RH = /~H(w = 0), however, some results o b t a i n e d for 2D systems at low doping and T -+ 0 indicate even RH < 0 [5], instead of the expected hole-like semiclassical be0921-4534/00/$ - see fi'ont matter (© 2000 Elsevier Science B.M Pll S0921-4534(00)00413-5
havior. Recently the l a t t e r result has been recovered analytically for a single hole doped into a 2D A F M [6]. Here we study the T = 0 Hall response as a ground state p r o p e r t y of a tight-binding model of strongly correlated electrons. We present two alternative ways to the d e t e r m i n a t i o n of the d.c. RH. One is by approaching the system with a ladder geometry, where a finite transverse width avoids the p r o b l e m of low-energy relaxation mechanism [7,8]. A n o t h e r is to investigate more general reactive Hall response in proper higherdimensional system, where again RH can be expressed with g r o u n d - s t a t e quantities only. This proves particularly meaningful approach to quasione dimensional systems, where the Hall constant appears to be related solely to the easy-axis stiffness. 2. L A D D E R
SYSTEMS
We first analyze a tight-binding model of interacting fermions on a ladder geometry with M legs in the y direction and L rungs in the x direction with periodic b o u n d a r y conditions [8]. The All rights reserved.
lq PrelovSek et al./Physica C 341-348 (2000) 129-132
130
following ingredients are needed in the model in order to study the Hall response: a) a finite transverse electric field $ y = A/e0, b) a homogeneous magnetic field B perpendicular to the ladder, introduced via the Peierls substitution into the hopping integrals. Within the Landau gauge A = B ( - y , 0) only the phases of hopping in the .r direction are modified as ~,, = (m - ~ ) ~ = (m - rh)e0B, c) a steady electric current density j* = j induced by piercing a closed ladder in the y direction with a flux, modifying the hopping term by a phase O. So the relevant terms of the model are
H = H~
Hy + Ha + Hint,
+
(1)
M
Hx = - t
Z rn=l
Z
~ i ( ~ - ° ) ' 3k C m , i + l , s C~m i s -H.c.),(2) -r
is
M-1
:-t'
*
z m=l
HA = A E ( m
+ H.c.),
(a)
is
-- fft)nrni,
(4)
im
where the interaction term Hint is not dependent on introduced fields. The idea is to study the ground state energy E of the system as a function of A, 0, ~ in order to evaluate the Hall constant RH, R. -
$ jB
A j~
(5)
In the absence of the magnetic field, ~ = 0, the starting point is an equilibrium and nonpolar ground state with the current density j c( OE/O0 = 0 and the polarization P c( OE/OA = 0. Such a ground state might correspond in finitesize ladder systems to 0o # 0, whereas it follows Ao = 0 by symmetry. To simulate the Hall effect we analyze systems with small current j # 0 imposed by a finite deviation 0 = 00 + 0, and with a magnetic field ~ # 0. At the same time we choose A # 0 so that the system remains nonpolar, P = 0. Evaluating A s induced by 0 and related j, we arrive at the result RH =
N/~°/,~ 0 o ' eo EA z~Eo o
(6)
E A0 : E o0a a E0
=
(7)
where only the derivatives of the ground state energy E(0, A, ~) evaluated at the equilibrium enter, and N = L M . It can be shown that the expression is equivalent to the one obtained via the linear response theory for dynamic RH(W) evaluated at w -----0,
-~H (w) =
I B
ayx (w)
(8) '
where aaZ(w) are dynamic conductivities. The new approach has nevertheless some clear advantages relative to the linear response theory approach: a) only the knowledge of only the ground state energy is required, b) the condition for j = 0, P = 0 in the reference ground state of finite size systems, being crucial in small systems, is more transparent. The method can be easily tested for noninteracting electrons on a two-leg (M = 2) ladder. It can be shown that expected semiclassical results are recovered: a) RH = --1/neeo for an empty upper band where ne is the density of electrons, and b) RH = 1/nheo for a filled lower band where nh = 2 - - ne is the density of holes in the upper band. As an open problem we study the Hall constant RH of correlated electrons within the isotropic t-J model, where
Hint=JE4 4,,
(9)
(iY) and J is equal along the rungs and legs, as well as t' = t. E(O, A, qo) are calculated in finite size ladders via the Lanczos diagonalization technique. A notrivial case now is already that of a single hole Nh = 1 on ladders with different M. Note that one would in this situation expect RH = N/eo, since we are deling with a single positively charged carrier (quasipartiele) in a system. Consistent although systematically higher value is found in M = 2 leg ladder for most J / t studied, while the semicalssical result seems to be approached in M > 2 ladders. It should be here pointed out that in finite-size s y s t e m s / ~ a v difo fers significantly from Eo~v, so that for J < t both quantities can even be of a different sign.
f~
Prelov~eket al./Plo,sica C 341-348 (2000) 129-132
20.0 i
.
i x
,
J=0,4t . - - RH*
.
o
10.0 ',,Xx x "..
c°~
O5X3 o6x3 x LSCO
o
x
0.0
-10.0
-20.0
131
easy to perform. Also, for D > 1 one would like to study systems with periodic boundary conditions where it is not possible to impose infinitesimally small homogeneous magnetic field B. An alternative is however to study the reactive w ~ 0 Hall response RH = /~H(W = 0) for a magnetic field modulated in the y direction, i.e. ~m = R e ( - i e ° B e i q m ) , q
0.0
0.2
0.4
0.6
0.8
n~
Figure 1. Hall constant eoRH vs. hole doping nh for the three-leg ladder and J = 0.4 t. b-~ll line serves as a guide to the eye. Dashed line represents the R~I result [3]. Shown are also experimental results for LSCO taken from Ref. [2].
(10)
Assuming a homogeneous current j* such a field induces a modulated electric field and potential in the y direction, i.e. we have to consider A --+ Aq. It should be noted that in such a situation at T = 0 the electric field $ y and E* are out of phase, as is the case of j* and g~, characteristic for a reactive response of as first considered for many-body systems by Kohn [9], At w ~ 0 we can identify,
2leg =
a~=(w ~ 0) = - - D o ,
(11)
O2
Results for more holes are less uniform. E.g. for two holes, N h 2, on a M = 2 ladder we get RH < 0 for a large J/t regime, consistent with previous studies [7]. This evident deviation from the semiclassical result seems to be related to the existence of the spin gap and the bound state of two holes on such a system. On the other hand, for odd leg (e.g. M = 3) ladder again results consistent with the semiclassical result are recovered. Moreover in M = 3 ladder we can follow the dependence of RH on hole doping nh with the result shown in Fig. 1. One can speculate that the M = 3 ladder already simulates well the behavior of a planar lattice. In fact our result agree quite reasonable with the experimental ones for RH in the normal state of prototype cuprate, La2_,Sr=CuO4 [2] (also presented in Fig. 1), where the planar hole concentration is directly related to x. =
3. REACTIVE HALL RESPONSE Previous approach cannot be extended directly to the 2D (or more D) systems, since the limit M -+ oo is ambiguous as well numerically not
aq~(w --~ 0) - ae~ Ofiq q20Aq '
a~x(~o--+ O) -
(12)
e2 a2E° q OAqO0
2Beg OD~ q2 OAq'
(13)
where fiq is the electron density modulated in the y direction, and D~, D~ are charge stifnesses (Drude weights), ^
( 0 ] - T~,qlO)
D 0~, q
--
2
-Re ~
-
(14)
(Oij~lrn)(rnlj~"[O),
m¢O
£rn
-- E0
defined for uniform and modulated currents respectively. Finally we get RH
=
--
10D~
q~o eOD~ Ofiq lira
~
10D~
e°D~ On
,
j~,q, (15)
assuming that q --+ 0 limit is smooth. The expression is appealing since it gives a direct, intuitive understanding of the change of sign of charge carriers in the vicinity of a metalinsulator transition. It e.g. immediately yields a semiclassical results both in the low-doping limit where D~ cx c~ and RH ~ 1/eoCh.
132
t? Prelovgek et al./Physica C 341-348 (2000) 129-132
4. 20
N
6
o N=2x7 * N=2x8 ,', N=2xl 0
10
......
o ....
6. 7.
o~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . •
O
•
@@
5.
O
8. -10
-20 0
.J/I=2. . . . . . . . 0.2
0.4
0.6
9. 0.8
Figure 2. RH vs. hole doping nh for weakly coupled t-J chains with J = 2t, calculated via the analytical relation (full line) and for various twoleg ladders with a interchain hopping t~/t = 0.5.
4. Q U A S I - 1 D S Y S T E M S The result (15) is particularly useful for a quasi1D system where D~ -~ D, i.e. the charge stiffness becomes the 1D easy axis one, since the weak interchain coupling reduces the coherence in the y-direction. We test the expression on the solvable 1D t-J model with J = 2t, where D(n) can be expressed analytically. Numerically we calculate R h for a M = 2 ladder, where now the coupling between legs is weak, i.e. only via the interchain hopping t ~ < t. Results shown in Fig. 2 seem to confirm the relation, although some scatter in numerical data is evident and can be attributed to finite-size effects. REFERENCES 1. For a review see e . g . N . P . Ong, in Physical
Properties of High Temperature Superconductors, ed. by D. M. Ginsberg (World Scientific, 2. 3.
Singapore, 1990), Vol. 2. H.Y. Hwang, et al., Phys. Rev. Lett. 72, 2636 (1994). H . E . Castillo and C. A. Balseiro, Phys. Rev. Lett. 68, 121 (1992).
B. S. Shastry, B. I. Shraiman, and R. R. P. Singh, Phys. Rev. Lett. 70, 2004 (1993). F . F . Assaad and M. Imada, Phys. Rev. Lett. 74, 3868 (1995). P. Prelov~ek, Phys. Rev. B 55, 9219 (1997). H. Tsunetsugu and M. Imada, J. Phys. Soc. Jpn. 66, 1876 (1997). P. Prelov~ek, M. Long, T. Marke~, and X. Zotos, Phys. Rev. Lett. 83, 2785 (1999). W. Kohn, Phys. Rev. 133, A171 (1964).